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UNITED STATES OF AMERICA. 



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ECLECTIC EDUCATIONAL SERIES 



ELEMENTARY 



MECHANICAL DRAWING 



FOE SCHOOL AND SHOP 



hi 



BY 

y 

FRANK ABORN, B. S. 

Drawing Master of Public Schools, Cleveland, Ohio 




I 

23 1886/? f 

-on 

'ASHING^- 



VAN ANTWERP, BRAGG & CO. 

CINCINNATI tf EW YORK 






ECLECTIC SYSTEM 
Of Freehand, Industrial, and Perspective Drawing 

COMPRISING S"K*\ C 

-*\ 3 <x \ J Introduction 
N V\ Price* 

1. Slate Exercises. Three months' course for youngest pupils. 

Per dozen sets $ .60 

2. Eclectic Drawing Books, Numbers I to IX. Carefully- 

graded and gradually progressive, from location and con- 
nection of points, to Exercises in Design, Mechanical Draw- 
ing, Perspective and Object Drawing, Historic Ornament, 

etc 

Numbers 1, 2, 3, per dozen 1.20 

Numbers 4, 5. per dozen 1.75 

Numbers 6, 7, 8, 9, per dozen 2.00 

(Practice Books, with full directions, per dozen .50 

Exercise Tablet No. 1, 7^ X 7%, 20 leaves, per dozen 1.20 

Exercise Tablet No. 2, 7>| X 9, 24 leaves, per dozen 1.75 

Exercise Tablet No. 3, 8 l / 2 X 10%, 24 leaves, per dozen 2.00 

4. Manual of Eclectic Drawing, free to teachers using this 

System. 

5. Elementary Mechanical Drawing. 121 pages .35 



"'Introduction price, for first introduction into schools where 
not already in use; and for single specimen copies, sent 
post-paid by mail, for examination with a view to first in- 
troduction. 

Supplies not for first introduction will be sent by express or 
freight on receipt of the Introduction price, or by mail, at 
one sixth added for postage and mailing. 

Van Antwerp, Bragg & Co., Cincinnati, New York, Boston. 



Copyright 

1886 

Van Antwerp, Bragg & Co. 



ECLECTIC PRESS 



PREFACE 



To the Teacher. 



In my endeavors to develop in the minds of 
pupils in grammar and high school classes a clear 
comprehension of the elementary principles of me- 
chanical drawing, it was found that something more 
was needed than the copying of a few isolated 
problems into drawing-books. The student gained 
a better grasp of this subject by solving a limited 
number of written problems, illustrating a principle 
or method of mechanical drawing that had been 
previously explained by the teacher from the black- 
board. 

The problems thus accumulated have been ar- 
ranged in the following pages with a view to econ- 
omizing the teacher's time as well as to furnish 
the student with facilities for independent work. 

To the Mechanic. 

It has been my aim in the following pages to 
illustrate only one new principle or method of 
mechanical drawing in the same problem, and to 

(3) 



6 PREFACE. 

make no more problems in any given case than are 
necessary to sufficiently illustrate the particular 
point under discussion. By so doing, I have been 
enabled to present short problems, and thereby to 
make each point of interest stand out in bold re- 
lief, and at the same time greatly to economize the 
time of the student. But this plan has prevented 
making the problems, as a whole, to apply to 
any particular trade or profession. This is no 
detriment to the work, however, although the 
individual who is seeking to learn that part of 
mechanical drawing which most especially applies 
to a particular trade may at first glance be disap- 
pointed, and think there is nothing here for him. 
To such let me say that the principles of mechan- 
ical drawing are every-where the same. Master 
these, and their application to ship-building, stair- 
building, architecture, sheet-metal work, etc., etc., is 
an easy matter. 

Frank Aborn. 

Cleveland, O., 
July, 1886. 



CONTENTS. 



Draughting Tools 



Part 1.— Geometrical Drawing. 



I. — Straight Lines 


. 


13 


IL— The Circle . 




15 


III. — Angles . 




26 


IV. — Triangles 




28 


Y. — Perpendicular Lines 


. 


31 


YI. — Tangent Lines 


. 


35 


VII. — Parallel Lines 


. 


36 


fill. — To Divide a Right Line into a Num- 




ber of Equal Parts 


37 


IX.— The Ellipse . 


. 


39 



Part II. — Constructive Drawing. 



Introduction 
I. — Scale Drawing 



(7) 



43 
45 



CONTENTS. 



CHAPTER. PAGE 

II.— Plans 48 

III.— Elevations 61 

IV. — Points in Inclined Surfaces ' . . 77 
V. — The Description of Points in a Sur- 
face of Regular Curvature . . 85 
VI. — The Intersection of Horizontal Lines 

with Inclined Surfaces ... 91 

VII.— Conic Sections 94 

VIII.— The Helix 96 

IX. — Section Drawing ..... 103 
X. — Foreshortened Dimensions . . . 106 
XI. — Development of Surfaces . . . 109 
XII. — Development of Oblique Conical Sur- 
faces 119 



ELEMENTARY 



MECHANICAL DRAWING. 



DRAUGHTING TOOLS. 



The Pencil. 

For Geometrical and Constructive drawing, the 
pencil must be of such quality and hardness that 
it will take and hold a fine point. 

The H grade Faber pencil is the best for school 
work. 

Sharpening the Pencil. — For ruled 
lines, the pencil should be sharpened to 
a flat edge (Fig. 1), as with the com- 
mon round point it is impossible to 
make the lines fine and of uniform 
width, which they must be if accuracy 
is to be secured. 

For free-hand lines, nothing will do 
but a keen, common, round point. 

As there are free-hand lines to be 
drawn, even in mechanical drawing, the 
student should have two pencils, one of which is 
sharpened to a flat, and the other to a round, point. 

0) 




Fig. 1. 



10 MECHANICAL DRAWING. 



The Scale. 



A Scale is a graduated ruler. 

For school work, a common box-wood ruler, with 
beveled edge, and graduated to sixteenths of an 
inch, will serve the purpose very well. 

Dividers. 

There is no tool that is so irredeemably bad and 
entirely worthless as the dividers that are cheap 
imitations of the better class of instruments. They 
look well to the uninitiated, but the set screws do 
not fit, the joints are rough and soon grind them- 
selves loose. Then, nothing will keep its place, 
and the instrument becomes unreliable and there- 
fore useless. But the Prang School Divider, made 
of sheet metal, and costing twenty-five cents, is 
thoroughly reliable in every way for school work. 
If, however, one wishes to have something better 

than this, there is noth- 
ing in the market, that 
it is economy to buy, at 
a less cost than three 
dollars; and a set of in- 
struments at this price 
should consist only of a pair of dividers, with 
needle-point, pencil-point, pen, and lengthening-bar 
(Fig. 2) ; and any set of instruments consisting of 
a greater number of pieces than this must of neces- 
sity be correspondingly higher in price or poorer in 
quality. 




DRAUGHTING TOOLS. 



11 



T-Square. 

A T-Square is an instrument, usually 
made of wood, consisting of a blade 
and head at right angles to each other 
(Fig. 3). 

For school purposes, a T-square should 
have a blade not more than twelve 
inches in length. 



Head 



Fig. 3. 




Tig. 4. 



Fig. 5. 



Set-Square. 

A Set-Square is an instru- 
ment in the form of a right- 
angled triangle (Figs. 4 and 5). 




Protractor. 

A Protractor is 

an instrument usu- 
ally in the form of 
a semicircular disk, 
having its periphery Fi s- 6 « 

graduated into 180 degrees (Fig. 6). 

For school work, a protractor made of horn, and 
three inches in diameter, is the best. 



Summary. 



Two H Faber pencils. 

One pair Prang School Dividers. 

One T-square, twelve inch blade. 

One set-square, four inches on longest edge. 



12 MECHANICAL DRAWING. 

One box-wood foot rule, beveled edge, graduated 
to sixteenths of an inch. 

One horn Protractor, three inches in diameter. 

One slate, eight inches by ten inches; or, draw- 
ing board, ten inches by twelve inches, and suitable 
paper and thumb tacks. 



PART I. 



GEOMETBICAL DRAWING. 

Geometrical Drawing is the description of lines 
arranged in conformity to some general rule called 
a geometrical law. 

Lilies are of two kinds, — straight and curved. 

Straight Lines are horizontal, vertical, and 
oblique, according to their direction with reference 
to the plane of the earth's surface. 

Curved Lines are either regular or irregular. 

A curve is Regular when its curvature follows 
an established law. 

A curve is Irregular when its curvature is not 
governed by any known rule or law. 



CHAPTER I.— Straight Lines. 
Section I. — Horizontal Lines. 

All level lines, i. <?., lines parallel to the plane of 
the earth's surface, are horizontal. > 

Horizontal Lines are drawn with the help of the 
T-square. 

Prob. 1. — Draw a horizontal line 4 in. long. 

(13) 



14 



MECHANICAL DRAWING. 





Fig. 7. 



Fig. 8. 



Solution. — Place the head of the T-square firmly 
against the left-hand edge of the slate or drawing 
board (Fig. 7,) and draw a line that is 4 in. in 
length along the edge of the blade. 

Prob. 2. — Draw a horizontal line 2- 1 in. long. 

Prob. 3.— Draw four horizontal lines 2>\ in. long 
and \ in. apart. 

Prob. 4. — Draw three horizontal lines, 1 in., 2 in., 
and 3 in. in length, and f in. apart. 

Prob. 5. — Draw six horizontal lines, \ in. apart 
and 2\ in. long, with their ends in a plumb line. 



Section II. — Vertical Lines. 

All plumb lines, i. e., lines that are perpendicular 
to the plane of the earth's surface, are vertical. 

Tertical Lines are represented with the aid of 
the T-square and set-square. 

Prob. 1. — Draw a vertical line 3-| in. long. 

Solution. — Place the head of the T-square firmly 
against the left-hand edsre of the slate or drawing 

£1 o o 

board, and while in this position set one of the 



GEOMETRICAL DRAWING. 15 

shorter edges of the set-square against it (Fig. 8.) 
Now, along the upright edge of the set-square, draw 
a line 3^- in. long, which will be the line required 
in the problem. 

Prob. 2. — Draw a vertical line 41 in. long. 

Prob. 3. — Draw two vertical lines 2f in. long and 
J in. apart. 

Prob. 4. — Draw" four vertical lines \ in., 1 in., 2 
in., and 3J in. long, and f in. apart. 



CHAPTER II.— The Circle. 
Section L — Definitions. 

A Circle is a plane figure bounded by a curved 
line, called its circumference, every part of which is 
equally distant from a point within it called its center. 

A Diameter of a circle is a straight line joining 
two points in the circumference, and passing through 
the center. Every circle may have an infinite num- 
ber of diameters. 

A Radius is a line extending from the center to 
the circumference. It is 
one half of a diameter. 

Prob. 1. — Describe a cir- 
cle -^2" m - m radius. 

Solution. — Set the divid- 
ers so that the distance 
between the needle-point 
and the pencil-point is -^ in. Place the needle- 




Fig. 9. Fig. 10. 



16 MECHANICAL DRAWING. 

point in the position of the center C, and holding 
the needle-point leg as nearly upright as possible, 
revolve the pencil leg about it, so that the pencil 
shall describe a continuous line ABD, every part of 
which is equally distant from its center C. 



Prob. 


2.— Describe a circle 1 in. radius. 


Prob. 


3. — Describe a circle ^ in. radius. 


Prob. 


4. — Describe a circle 1^ in. radius. 


Prob. 


5. — Describe a circle -^ in. radius. 


Prob. 


6. — Describe a circle 1^ in. diameter. 


Prob. 


7. — Describe a circle 1 in. diameter. 


Prob. 


8. — Describe a circle 1-J- in. diameter. 


Prob. 


9. — Describe a circle If in. diameter. 


Prob. 


10. — Describe a circle ^g 5 - in. diameter. 



Section II — Arcs of Circles. 

Any portion of a circumference less than the 
whole is called an arc. 

Every circumference is considered as consisting 
of 360 equal arcs. 

Each of these 360 arcs is called an arc of 1 de- 
gree. 

The name of an arc depends upon the number of 
degrees that it contains. 

One fourth of a circumference is an arc of 90 
degrees, and is written 90°. 

One third of a circumference is an arc of 120 
degrees, and is written 120°. 

Three fourths of a circumference is an arc of 270 
degrees, and is written 270°. 



GEO ME TRICA L DBA WING. 



17 



One half of a circumference is an arc of 180 de- 
grees, and is written 180°, etc., etc. 

A Protractor used 
in school work is 
usually a semicir- 
cular disk, and 
-therefore its arc 
contains 180°. Usu- 
ally these degrees 
are marked in two 
lines. One of these 
lines gives the 
number of degrees, 
counting from the 
left-hand end of the diameter, and one gives the 
number of degrees counting from the right-hand 
end of the diameter. 




Fig. 11. 



Prob. 1. — Describe an arc of 60°, with a radius of 



in. 



/b 



Solution. — Describe a cir- 
cle ABD, | in. in radius. 
Draw a diameter AD. Place 
the protractor on the circle 
ABD, so that its center and 
diameter coincide with the 
center and diameter of the 
circle, and mark the 60° 
point, 6, at its edge. (Fig. 
11.) Remove the protractor, 
and draw the line bBC to the center. (Pig. 12.) 
The point B, where the line crosses the circumfer- 
ence, will be one end of the required arc; the 




Fig. 12. 



18 



MECHANICAL DRA WING. 



other end is at D, the end of the diameter from 
which it was measured. BD is the required arc. 

Prob. 2. — Describe an arc of 120°. Radius, -| in. 
Prob. 3. — Describe an arc of 90°. Radius, 1-J- in. 
Prob. 4. — Describe an arc of ,30°. Radius, If in. 
Prob. 5. — Describe an arc of 270°. Radius, 1^- in. 



Prob. 6. — Describe an arc of 45°. Radius 



> 32 



111. 



Section III. — To (/escribe an arc with a given radius 
that will be equal in length to the circumference of a 
given circle. 

The length of a degree varies with the length of 
the radius of the circle. 

Prob. 1. — With a radius of | in., describe an arc 
that will be equal in length to the circumference of 
a circle having a radius of J in. 

Solution. — To find what part of the circum- 
ference of the larger 
circle is equal to the 
whole circumference of 
the smaller, we divide 
the radius of the smaller 
circle hy the radius of 

3 

the larger : !=-§-. Hence, 
Fig. 13. 8 I 1 ' 

an arc which is f of 
the circumference of a circle whose radius is f in. 
will be equal to the entire circumference of a circle 




ha vine: a radius of 



in. 



As there are 360° in an 



entire circumference, \ of it would be \ of 360° = 
51.43°, and f of it would be three times 51.43° = 



GEOMETRICAL DRAWING. 19 

154.29°. Describe an arc, ADJB, having a radius of 
| in., and on it lay off AB containing 154.29°. 
This arc will be equal in length to the circle having 
a radius of f in. (Fig. 13.) 

Prob. 2. — With a radius of 1\ in., describe an arc 
of a circle which is equal to the circumference of a 
circle of J in. radius. 

Prob. 3. — With a radius of 2\ in., describe an arc 
of a circle which is equal to the circumference of a 
circle of J in. radius. 

Prob. 4. — Describe an arc of a circle having a 
radius of 1\ in., which is equal to the circumfer- 
ence of a circle, radius If in. 

Prob. 5. — Describe an arc, radius If in., equal to 
the circumference of a circle, radius 1^ in. 

Prob. 6. — Describe an arc, radius If in., equal to 
the circumference of a circle, radius -^ in. 

Eule. — Divide the radius of the given circle by 
the radius of the required arc. Reduce the result- 
ing fraction to degrees, and lay off the number of 
degrees thus found on the circumference of a circle 
having the radius of the required arc. The arc so 
laid off will be equal to the circumference of the 
given circle. 

Section IV. — To find the Circumference of a Circle. 

The Circumference of a circle is equal to the 
product of the diameter multiplied by 3.1416. 

Note. — The exact ratio between the diameter and the cir- 
cumference of a circle can not be given in figures, but 3.1416 
is near enough for all ordinary purposes. 
M. D.-2. 



20 MECHANICAL DRAWING. 

Prob. 1. — What is the length of the circumfer- 
ence of a circle that is 11 in. in diameter? 

Solution. — As the diameter is 1.5 in., the length 
of the circumference must be 3.1416 times 1.5 in., 
which equals 4.7124 in. Hence, 4.7124 in. is the 
length of a circle 1J in. in diameter. 



Prob. 2. — What is the length of the circumfer- 
ence of a circle having a diameter of 2 in. ? 

Prob. 3. — What is the length of the circumfer- 
ence of a circle 41 in. radius ? 

Prob. 4. — What is the diameter of a circle having 
a circumference of 6.2832 in.? 

Prob. 5. — What is the length of the circumference 
of a circle having a radius of 3 in.? 

Prob. 6. — What is the radius of a circle having a 
circumference of 10 in. ? 

Prob. 7. — Describe a circle having a circumference 
of 5.4978 in. 

Prob. 8. — Describe a semi-circumference 2.7489 in. 
long. 

Prob. 9. — Describe a quarter circumference 7.854 
in. long. 

Prob. 10. — Describe an arc of 45°, 5.8901 in. long. 



Section V. — Bisection of Circular Arcs. 

To bisect an arc is to divide it into two equal 
parts. 

Prob. 1. — Bisect an arc of 60°; radius, ^-| in. 



GEOMETRICAL DRAWING. 21 

Solution. — Describe a circle if in. in radius, and 
on its circumference lay off an arc of 60°, AB. With 

A and B as centers, describe ^ _ A 

arcs of equal radii intersecting 

in b and b' . Join b and b r by / ,v 

a straight line, aud this line, \ 

where it cuts AB in a, bisects 

the arc AB. Aa and a^ are ^—'' 

equal. lg ' 

Prob. 2. — Bisect an arc of 90°, 1J in. radius. 
Prob. 3. — Bisect a semi-circumference } in. radius. 
Prob. 4. — Divide an arc of 120° into four equal 
arcs. Diameter, 1| in. 

Prob. 5. — Divide an arc of 
/^I^>X 170°, 1J in. radius, into eight 

// ^\\ equal arcs. 

If \\ Prob. 6. — Represent the face 

"" of a semicircular arch consisting 

of eight equal blocks. Radius 
of inner circle, 1J in., and radius of outer circle, 
2 in. (Fig. 15.) 

Prob. 7. — An arch is 4 in. in s^V^lh^ 

diameter on the inner circle. /S^ ^\\ 

The key-stone is 20° wide and by \\ 

-J in. long measured on the /_] r □ 

radius, and on each side of the Fi s- 16 - 

key are eight equal blocks f in. long measured on 

the radii. Represent the face of the arch. (Fig. 16.) 

Section VI. — Trisection of Circles. 

To trisect a circle is to divide it into three equal 
arcs. 



22 



MECHANICAL DRAWING. 



A Chord of an arc is a line joining its extremi- 
ties. (See Fig. 17.) 




Fig. 17. 



The chord of an arc of 60° is equal to the radius 
of the arc. 

Prob. 1. — Draw a circle f in. in diameter, and 
divide its circumference into six equal arcs. 

Solution. — Describe a circle AF, f in. in diam- 
eter, and, without changing the di- 
viders, set the needle-point at any 
point in the circumference, A, and 
describe arcs, cutting the circumfer- 
ence in B and F. Set the needle-point 
at F and B, and describe arcs cut- 
ting the circumference in E and C. 
So proceed, and the last arcs will meet in D, and 
the circumference will be divided into six equal 
arcs. The chord of any of these arcs is equal to the 
radius of the circle. 

To divide the circumference of a circle into three 
equal divisions: Divide the circumference into six 
equal arcs, and two of these arcs will constitute one 
third of the whole circumference. 

To divide the circumference of a circle into twelve 
equcd divisions: Divide the circumference into six 




GEOMETRICAL DRAWING. 23 

equal arcs, and then divide each of these into two 
equal arcs. 

Prob. 2. — Describe two circles, 1J in. and f in. in 
diameter, on the same center, and divide the annular 
space between them into twelve equal parts. 

Prob. 3. — Describe a semi-circumference, diameter 
2f in., and divide it into three equal arc3. 

Prob. 4. — Describe a circle ; divide its circumference 
into three equal arcs, and join the division points. 
The inscribed figure will be an equilateral triangle. 

Prob. 5. — Describe a circle; divide its circumfer- 
ence into six equal arcs, and join the division 
points. The inscribed figure will be a hexagon 
(regular six sided polygon). 

Section VII. — Concentric or Parallel Circles. 

Circles are concentric when they have a common 
center : a, b, d, and e are concentric 
circles because they have the com- 
mon center C. 




Note. — In describing circles, there is a 
strong temptation to incline the dividers 
to one side. In doing this, the center of 
the circle is so torn and enlarged that it 

can not be used again. It becomes necessary, therefore, for 
the student to practice making concentric circles until he has 
learned to hold the dividers erect. 

Prob. 1. — Describe six concentric circles, J in. 
apart, the radius of the largest circle being 2 in. 

Prob. 2. — Describe six concentric circles, } in., 1 
in., 1J in., 2 in., 2| in. in radius. 



24 MECHANICAL DRAWING. 

Prob. 3. — Describe three concentric circles, 3J in., 
3 J in., and 3 in. in diameter. 

Prob. 4. — Describe four concentric circles, ^g in., 
-j^- in., \^ in., and f| in. in radius. 

Prob. 5. — Represent the top of your ink well. 

Prob. 6. — Describe two concentric circles, 2 in. and 
2J in. in diameter. Divide the circumference of the 
smaller one into six equal parts, and with each of 
the division points as a center, describe six pairs of 
concentric circles, similar and equal to the first. 



Section VI II.— Tangent Circles. 

Circles are tangent when their circumferences 
touch each other in one point only. Circles may 
be tangent externally or internally. 

Externally Tangent Circles. — Circles are exter- 
nally tangent when the distance between their cen- 
ters is equal to the sum of their radii. 

Prob. 1. — Describe two circles externally tangent 
to each other; radii, ^ in. and -^ in. 

Solution. — As these circles are externally tan- 
gent, the distance between their 
centers will be equal to the sum 
of their radii : T 3 g in.-f- ^ in. = -J-J 
in. Draw a line CO J-f in. long 
(Fig. 20.) With Cas a center, and 
with a radius of T 3 g in., describe 
Fig - 20 - the circle A ; with O as a center, 

and with a radius of ^ in., describe the circle B. 

These circles, A and B, will be externally tangent at b. 




GEOMETRICAL DRAWING. 25 

Prob. 2. — Describe two externally tangent circles ; 
radii, J in. and J in. 

Prob. 3. — Describe two externally tangent circles ; 
radii, ^f in. and f in. 

Prob. 4. — Describe two externally tangent circles ; 
the sum of the radii to be 3 in., and one radius 
to be twice as long as the other. 

Prob. 5. — Describe two concentric circles, J in. and 
f in. in radius, externally tangent to two similar and 
equal concentric circles, with the larger of one pair 
of circles tangent to the smaller of the other 
pair. 

Rule.— Draw a line equal in length to the sum 
of the given radii, and, with the ends of this line as 
centers, describe the given circles. The circles thus 
described will be externally tangent, and the point 
of contact will be on the line joining their centers. 

Internally Tangent Circles.— Circles are inter- 
nally tangent when the distance 
between their centers is equal 
to the difference of their radii. 

Prob. 1. — Draw two internally 
tangent circles ; radii, -^ in 
and -^ in. 

Solution. — Since ^ — tq = 
^-, the distance between the 

centers of the given circles is T 6 g- in. Draw a line, 
Cc f (Fig. 21,) T 6 g- in. in length, and with c' and C as 
centers, describe two circles, abd and ABD, -^ in. 
and a>_ i n . radii. ABD and abd are internally tan- 
gent at a. 




26 MECHANICAL DRAWING. 

Prob. 2. — Describe two internally tangent circles, 
1 in. and | in. radii. 

Prob. 3. — Describe two internally tangent circles, 
If in. and 1J in. in diameter. 

Prob. 4. — Describe two arcs of 90°, T 9 g in. and -J in. 
radii, internally tangent at one end, and two other 
similar arcs similarly tangent, having a common 
point of contact, and all the centers on the same 
right line. 

Rule. — Draw a line equal to the difference of the 
radii of the tangent circles, and with the ends of 
this line as centers describe circles with the given 
radii. The circles so drawn will be internally tan- 
gent to each other. 



CHAPTER III.— Angles. 

An angle is formed by the meeting of any two 
lines. (See Fig. 22.) 



A 

Fig. 22. 



The point of meeting of two lines is called the 
apex of the angle. 

If the apex of an angle be made the center of a 
circle, the arc intercepted by the sides of the angle 
is said to subtend the angle. 



GEOMETRICAL DRAWING. 



27 



An angle is measured by the subtending arc. (See 
(Fig. 23). 





-?»6, 



-% 




Fig. 24. 

and 1J in. in 



Fig. 23. 

'Prob. 1. — Draw an angle of 30°. 

Solution. — With any radius de- 
scribe a circle ABB (Fig. 24). 
Lay off on this circle an arc of 30°, 
AB. From the center, 0, draw \ 
radial lines through A and B. The 
arc AB, 30°, subtends and there- 
fore measures the angle ACB. 

Prob, 2. — Draw two lines, 1 in. 
length, forming an angle of 45°. 

Prob. 3. — Draw three lines, 2 in. in length, meet- 
ing in a common point, and forming three angles of 
120° each. 

Prob. 4. — Draw three lines, 1J in. in length, meet- 
ing in a common point, and forming three angles of 
210°, 60°, and 90°. 

Prob. 5. — Draw a horizontal line, 2 in. long, 
bisecting two lines 1J in. in length, one of which 
makes an angle of 45°, and the other an angle of 
135°, with the connecting line. 

Rule. — Describe an arc containing the number of 
degrees in the required angle. Draw the radii at 
the ends of this arc. The angle formed by the 
radii thus drawn will be the angle required. 

M. D.— 3. 



28 MECHANICAL DRAWING. 



CHAPTER IV.-Triangles. 

A Polygon is a plane figure having many sides 
and an equal number of angles. 

A Triangle is a polygon that has three angles. 

Section I. — To draw a triangle token its three sides 
are given. 

Prob. 1. — Draw a triangle, the sides of which are 
•|| in., !~J in., and -^ in. in length. 

Solution.— Draw a line, AB, -|| in. in length, for 
one of the sides of the required 
triangle. With one of the ends 
of this line, A, as a center, describe 
an arc, ac, T 7 g in. in radius, and with 
Flg ' 25 ' the other end of the line, B, as a 

center, describe an arc fj in. radius, bd, cutting ac 
in C. Draw AC and BC, and the figure thus 
formed, ABC, is the required triangle, of which the 
sides are -|f in., |~J in., and -^ in. (Fig. 25.) 

Prob. 2.— Draw a triangle, the sides of which are 
1J in., 1J in., and 1 in. long. 

Prob. 3.— Draw a triangle, the sides of which are 
2 in., | in., and If in. long. 

Prob. 4. — Draw an equilateral triangle, the sides 
of which are 1J in. in length. 

Knle.— Draw a line equal in length to one of the 
given sides of the required triangle, and, with the 
ends of this line as centers, describe intersecting 




GEOMETRICAL DRAWING. 29 

arcs, the radii of which are equal to the other sides. 
From the point of intersection of these arcs, draw 
lines to the ends of the line already drawn. The 
three lines thus drawn will be the three sides of the 
required triangle. 

Section II. — To draw a triangle when one angle and 
two sides are given. 

Prob. 1. — Draw a triangle, of which one angle is 
30°, and two of the sides are -^ in. and 1-^- in. in 
length. 

Solution.— Draw any line AB, l T 3 g in. in length 
(Fig. 26.) At A draw a line 
AC, -fg in. in length, and 
forming an angle of 30° with 
AB. Draw OB. The triangle Fig. 26. 

ABC, thus formed, is the required triangle. 

Prob. 2. — Draw a triangle, of which one angle is 
45°, and two sides are 2 in. and If in. in length. 

Prob. 3. — Draw a triangle, of which one angle is 
90°, and two sides are 3 in. and 2J in. in length. 

Prob. 4. — Draw a triangle, of which one angle is 
60°, and two of the sides are If in. and if in. in 
length. 

Rule. — Draw two lines, equal in length to the 
given sides of the required triangle, and forming an 
angle equal to the given angle. Join the ends of 
these lines by a right line, and the three lines thus 
drawn will form the required triangle. 





30 MECHANICAL DRAWING. 



Section III. — To draw a triangle having given one side 
and two angles. 

Prob. 1. — Draw a triangle, one side of which is 
l T 5 g- in. and two angles of which are 30° and 45°. 

Solution. — Draw a line, AB (Fig. 27), l T 6 g- in. in 

length, for one of the sides 

of the required triangle. 

At A draw A C, making an 

angle of 45° with AB, and 

at B draw BC, making an 

Fig ' 27 * angle of 30° with AB. 

Produce these lines until they meet. Then ABC, 

having one side 1^- in. in length, and two angles 

of 30° and 45° is the required triangle. 

Prob. 2. — Draw a triangle, of which one side is 1J 
in. in length, and two angles are 45° and 90°. 

Prob. 3. — Draw a triangle, of which one side is 1 
in., and two angles are 60° each. 

Prob. 4. — Draw a triangle, of which one side is 
2 J in. in length, and two angles are 30° and 22 J°. 

Prob. 5. — Draw a triangle, of which one side is f 
in. in length, and two angles are 75° each. 

Rule. — Draw a line equal in length to the given 
side of the required triangle. At the ends of this 
line, and on the same side, draw lines forming the 
given angles of the required triangle. Produce these 
lines until they meet. The figure formed by the 
three lines thus drawn will be the required triangle. 



GEOMETRICAL DRAWING. 31 



CHAPTER V. — Perpendicular Lines. 

Two lines are perpendicular to each other when 
the angles at their point of meeting are each 90°. 

Section I. — To draw a line perpendicular to another 
line at a point in the middle of the given line. 

Prob. 1. — Draw a line 1 in. long, and erect a per- 
pendicular to it at a point midway between the 
ends. 

Solution. — Draw a line, AB (Fig. 28), 1 in. in 
length. With A and B as cen- 
ters, and the same radius, de- 
scribe two arcs of circles, cutting 

each other in b and b' on both A 

sides of the line AB. Join b 
and b f , and the line bb' will be 
perpendicular to the line AB 
and cut it in the middle. Fig> 2 8. 

Prob. 2. — Draw a line 2f in. in length, and cross 
it in the middle by a line perpendicular to it. The 
cross line to be If in. in length and equally divided 
by the first line. 

Rule. — Draw a line equal to the given line, and 
with the ends of this line as centers, describe two 
arcs, with the same radius, cutting each other on 
both sides of the given line. Join the points of in- 
tersection of these arcs, and the line thus drawn 
will be perpendicular to the given line at a point in 
the middle. 



32 MECHANICAL DRAWING. 

Section II. — To draw a line perpendicular to another 
at a point not in the middle of the given line. 

Prob. 1. — Draw a line 1J in. in length, cut by a 
line perpendicular to it at a point J inch from the 
right-hand end of the given line. 

Solution.— Draw a line AB (Fig. 29), 1| in. in 

length. Locate a point, a, 

1 in. from its right-hand end. 

^ With a as a center, describe 

; b ' arcs of equal radii, cutting 

AB in h and V. With b and 

.*■*%? b' as centers, describe arcs of 

Fig - 29 - equal radii cutting each other 

on both sides of AB, in c and c' . Join c and c' ? and 

the line cc' is perpendicular to AB at a point a, J in. 

from the right-hand end of AB. 

Prob. 2. — A line If in. in length is cut by another 
line perpendicular to it at a point -J in. from the 
left-hand end. Draw the lines. 

Prob. 3. — Draw a line 3 in. long, crossed If in. 
from its left-hand end by a perpendicular line 2 in. 
long, which it bisects. 

Rule. — Draw a line equal to the given line, and 
in it locate the given point. With this point as a 
center, describe arcs of equal radii cutting the 
given line on each side of the given point. With 
these points as centers, describe arcs of equal radii 
cutting each other on both sides of the given line. 



GEOMETRICAL DRAWING. 



33 



A line joining the points of intersection of these arcs 
will pass through the given point, perpendicular to 
the given line. 

Section III. — To draw a line 'perpendicular to a given 
line from a point not in the given line. 



I- 7 

- L n 




Fig. 30. 



Prob. 1.— Draw a triangle, the sides of which are 
in., l^g- in., and f in. long, having a line per- 
pendicular to its longest side drawn from the angle 
opposite it. 

Solution.— Draw the given triangle, ABC. (Fig. 
30). With C as a center, 
describe an arc cutting 
the longest side, AB, in 
two points, b and b' . 
With these points as cen- 
ters, describe arcs of equal 
radii cutting each other 
on both sides of AB, in c and c'. From C draw a 
line through c and c' . This line will be the re- 
quired perpendicular. 

Prob. 2. — Draw a line If in. long, perpendicular 
to another line 2 in. long, from a point that is 1J 
in. from the left-hand, and If in. from the right- 
hand, end of the given line. 

Rule. — Draw the given line, and locate the given 
point. With this point as a center, describe an 
arc cutting the given line in two points. With 
these points as centers, describe arcs of equal radii 



34 



MECHANICAL DRAWING. 



cutting each other on both sides of the given line. 
From the given point, draw a line through the 
points of intersection of these arcs, and this line 
will be the required perpendicular. 







F 






']•' 




,'C N 


f**N^ 


/ b,' 




R 'i 



Section IV. — To draw a line perpendicular to, and at 
the end of, a given line. 

Prob. 1. — The sides of a right-angled triangle ad- 
jacent to the right angle are 1 in. and J in. in 
length. Draw the triangle. 

Solution. — Draw a line AB (Fig. 31), 1 in. in 

length. With B as a 
center, describe a semi- 
circle, heed. Divide this 
semicircle into three 
equal arcs, be, ce, and 
ed. Bisect the middle 
arc, ce, in /, and draw 
Bf. Extend Bf to F, making BF J in. in length. 
Join A and F, and FAB is the required right- 
angled triangle. 

Prob. 2. — Draw a square; sides, 1\ in. in length. 

Prob. 3. — Draw a square; sides, 1^- in., making 
angles of 30° and 60° with the horizon. 

Prob. 4. — Draw a rectangle; sides, 1\ in. and 1^ 
in. in length. 

Rule. — Draw the given line, and, with one end of 
this line as a center, describe a semicircle, of which 
the given side produced is the diameter. Divide 



Fig. 31. 



GEOMETRICAL DRAWING. 35 

this semicircle into three equal arcs. Bisect the 
middle one of these arcs, and through the bisecting 
point draw a line to the end of the given line. 
This line will be the required perpendicular. 



CHAPTER VI.— Tangent Lines. 

A line is Tangent to a circle when it touches it 
at one point only. 

The point where the line meets the circle is called 
the point of tangency or point of contact. 

A tangent line is always perpendicular to the 
radius at the point of contact. 

Prob. 1. — Draw a circle J in. in diameter, and a 
line 1 in. long tangent to the circle; the point of 
contact to be in the middle of the line. 

Solution. — Draw the circle AB (Fig. 32), J in. in 
diameter. Draw the radius CA. 
Draw the line DE, perpendicular to 
CA, at the end of it, and J in. on 
each side of the point of contact, A. 
AB is the required circle, and DE is 
the required tangent to the circle. j> 

Fig. 32. 

Prob. 2. — Draw a circle If in. in diameter, and a 
tangent line 1 in. long; point of contact in the 
middle of the line. 

Prob. 3. — Draw a circle J in. radius, and a tan- 
gent line 1^ in. long; point of contact at the end 
of the tangent. 




36 MECHANICAL DRAWING. 

Prob. 4. — Describe two concentric circles f in. 
and li in. diameter, and draw a tangent line to each 
of them 2 in. in length ; points of contact on the 
same radius, and Jin. from the ends of the tangents. 

Prob. 5. — Describe two circles 1 in. and J in. in 
diameter, 2 in. between centers, and one line tan- 
gent to both of them. 

Rule. — Describe the given circle, and draw the 
radius at the point of tangency. Erect a perpen- 
dicular at the end of this radius, and the line so 
drawn will be tangent to the circle. 



CHAPTER VII.— Parallel Lines. 

Two lines are Parallel when the distance be- 
tween them remains the same throughout their 
entire length. 

Prob. 1. — Draw two parallel lines 1J in. in length 
and i in. apart. 

Solution. — Draw a line AB (Fig. 33), 1J in. in 
c length. As the parallel line 

~^T~^Y' iT"V i s i m * from this line, with 

A B any two points on the line 

AB, as centers, describe two 
arcs of J in. radius on the same side of AB. Draw 
a line CD, 1J in- long? tangent to these arcs, and 
this line will be parallel to AB and i in. from it. 



GEOMETRICAL DRAWING. 37 

Prob. 2. — Draw three pairs of parallel lines: 3 in. 
long and | in. apart; 5 in. long and f in. apart; 
and \\ in. long and f in. apart. 

Prob. 3. — Draw three sets of three parallel lines 
each: 3 \ in. long, \ in. apart; 5J in. long, J in. 
apart ; 4J in. long, -^ in. apart. 

Prob. 4. — Draw three sets of four parallel lines: 
3 in. long, -^ i n . apart; 4J in. long, % in. apart; 
5 in. long, T 7 g- in. apart. 

Prob. 5. — Draw the following described lines: first 
line, If in. long; second line, 2f in. long, parallel 
with first line, and f in. from it; third line, 3 J in. 
long, parallel with second line, and -^ in. from it; 
fourth line, 3 J in. long, parallel with third line, and 
^ in. from it; fifth line, 4f in. long, parallel with 
fourth line, and T 3 ¥ in. from it. 

Rule. — Draw the given line, and, with two points 
in this line as centers, describe two arcs of equal 
radii on the same side of it. Draw a line tangent 
to these arcs, and this line will be parallel with 
the given line. 



CHAPTER VIII.— To Divide a Right Line into 
any Number of Equal Parts. 

Prob. 1. — Divide a line 1^ in. long into five equal 
parts. 



38 



MECHANICAL DRAWING. 



v: 




Fig. 34. 



Solution. — Draw a line 0-0 (Fig. 34), 1J in. long. 
With and as centers, draw 
arcs aa! and bb' with equal radii 
on opposite sides of 0-0. From 
draw the two lines 0-5 tangent to 
these arcs. Beginning at 0, lay 
off five equal divisions of any 
length on each of these lines, and 
join 0-5, 1-4, 2-3, 4-1, and 5-0. 
These lines will divide the given 
line, 0-0, into five equal parts. 

Prob. 2. — Divide a line 4^- in. long into four 
equal parts. 

Prob. 3. — Divide a line 3|f in. long into seven 
equal parts. 

Prob. 4. — Divide a line 2^ in. long into eleven 
equal parts. 

Prob. 5. — Draw a line 3 in. long, and divide it 
into nine equal parts. 

Prob. 6. — Draw a 4 J in. square, and divide it into 
nine smaller squares. 

Prob. 7. — Draw a circle 3J in. in diameter, crossed 
by four parallel lines perpendicular to its vertical 
diameter, and dividing it into five equal parts. 

Prob. 8. — The sides of a triangle are | in., 1J in., 
and 2J in., and it is crossed by three lines parallel 
with its longest side and dividing the other sides 
into four equal parts. Draw the triangle and the 
lines crossing it. 

Rule. — Draw the given line, and with the ends of 
this line as centers, describe arcs of equal radii on% 
opposite sides of the given line. From the ends 



GEOMETRICAL DRAWING. 39 

of this line, draw two lines tangent to these ares, 
and lay off on them the required number of equal 
divisions, beginning at the end of the given line. 
Join the corresponding division points, and the lines 
so drawn will divide the given line into the required 
number of equal parts. 



CHAPTER IX.— The Ellipse. 

An Ellipse is a curved line that has two centers, 
called its foci, and two diameters, called its major 
and minor axes. 

The Major Axis is the longest diameter of the 
ellipse. 

The Minor Axis is the shortest diameter of the 
ellipse. 

The major and minor axes of the ellipse are per- 
pendicular to each other. 

The sum of the distances of any point on the 
curve from the foci is equal to the major axis of 
the ellipse. 

To Draw an Ellipse. — First Method. 

Prob. 1. — Draw an ellipse, the axes of which are 
J in. and 2 in. 

Solution. — For the major and minor axes of the 
required ellipse, draw AB and DC, 2 in. and -J in. 
in length, perpendicular to and bisecting each other. 
With the center at (7, and the radius equal to one 
half of AB, describe arcs cutting AB in F and F r . 



40 



MECHANICAL DRA WING. 




Then, in each case eF 
In this way take as 



eF' 



These points, F and F, are the foci of the required 
ellipse. To find a point through which an ellipse 
drawn on these diameters would pass, locate any 

point, F, on AB, 
and with AE and 
EB as radii, de- 
scribe arcs from 
each of the foci 
as centers, cut- 
ting each other 
in e, e, e, and e. 
AB, the major axis, 
many points as necessary 
on AB, and proceed as before. When a sufficient 
number of points has beeu thus established, draw 
a curved line through them. This line will de- 
scribe the required ellipse, Aeg, Cge, Beg, Dge. 
(Fig. 35.) 

Prob. 2. — Draw an ellipse, of which the axes are 
1J in. and 2J in. 

Prob. 3. — Draw an ellipse, of which the axes are 
f in. and 1J in. 

Prob. 4. — Draw an ellipse, of which the axes are 
J in. and If in., and another on the same axes 1 in. 
X2i in. 

Prob. 5. — Draw two ellipses on the same diame- 
ters : J in. X If in. and 1 in. X 3J in. 

Prob. 6. — Draw an ellipse If in. X 2 in. 

Prob. 7. — Draw two ellipses on the same diame- 
ters. The minor axis of one of them is | in., and 
of the other If in. The major axis in each is twice 
as great as the minor axis. 



GEOMETRICAL DRAWING. 



41 



Rule. — Draw the major and the minor axes. Lo- 
cate the foci. Divide the major axis into two un- 
equal divisions. With each of the parts of the 
major axis thus obtained as radii, and the foci as 
centers, describe arcs intersecting in each quadrant 
of the ellipse. Make as many divisions of the 
major axis, and proceed as before, as may be neces- 
sary to determine the curve sufficiently. When all 
the points have been de- 
termined, draw a line join- 
ing them and the ends of A ^r 

the major and minor axes. 
The line so described will 
be the required ellipse. « 




Second Method.— To De- 
scribe an Ellipse with a 
String. 

, Note. — To describe an ellipse 
with a string is frequently very 
convenient; but as a string is 
elastic, it is unreliable. For 
this reason, it should never 
be used in describing ellipses, 
except where only quick, ap- 
proximate results are required. 

Prob. 1. — Draw an ellipse 
T \ in. X If in. 



Solution. — Draw the 
axes of the ellipse, AB 
and CD (Fig. 36). With C as a center, and AE as 
a radius, locate the foci F and F 1 . Double a 




42 MECHANICAL DRAWING. 

string and tie a loop, the length of which, ab, is 
equal to AB. Set a pin in each focus, F and F, 
and drop the loop over them, xyz. Place the pencil 
against the loop, and push it out to C. Then move 
the pencil either way, pressing against the string, 
and describe one half of the ellipse. In the same 
way, push the loop to D and repeat. The two lines 
so described will form an approximate ellipse. 

Prob. 2. — Draw an ellipse ; axes, 1J in. and 2J in. 
Prob. 3. — Draw an ellipse ; axes, 1 in. and 1 J in. 
Prob. 4. — Draw an ellipse; axes, 1J in. and If in. 
Prob. 5. — Draw an ellipse; axes, J in. and 2 in. 



PART II. 



OOUSTEUOTIVE DRAWING. 

INTRODUCTION. 

Art aims so to present ideas as to elevate and 
develop sentiment and feeling. In other words, art 
is emotional, and therefore variable. Hence, any 
work may be excellent art, and still it may not be 
exactly and impartially true. 

Science is matter-of-fact, practical, and therefore 
exact. It aims to state or describe only facts in 
the simplest and most unmistakable way. 

All solids — objects that occupy space — have size ; 
that is, every part of a solid has its own length, 
breadth, and height, called its dimensions. 

The Dimensions of a solid may be described by 
two very different kinds of drawing, Pictorial 
Drawing and Constructive Drawing. 

Pictorial Drawing is the art of describing the 
appearance of the dimensions of a solid on a flat 
surface. 

Constructive Drawing is the science of describ- 
ing the exact dimensions of a solid on a flat surface. 

M. D.— 4. (43) 



44 



MECHANICAL DRAWING. 




Fig. 37. 



Every science has its exact and invariable 
methods, and the method of ascertaining the dimen- 
sions of a solid that is to be described by con- 
structive drawing may be explained as follows: 
On a common business card draw two lines, ab 
and cd (Fig. 37), perpendicular to 
each other. Through their point 
of intersection, e, thrust a straight 
common pin perpendicular to ab 
and cd. When the card lies on 
some level surface, so that ab and 
cd are horizontal, its length and breadth at a and c 
are measured on ab and cd, and the height or thick- 
ness is measured on the line cf. 

If the card should stand so that ef and 
cd are horizontal (Fig. 38), the length 
would be measured on cd, and the breadth 
or thickness on fe, and the height on ab. 
When the card stands so that ef and 
ab are horizontal (Fig. 39), the length 
and breadth or thickness are measured 
on the lines ef and ab, and the height 
is measured on the line cd. 

If the card should be inclined, and 
onlv cd remain 
horizontal (Fig. 
39a), perhaps like 
the side of a gable 
roof, then the length would 
be measured on cd; but as ef 
would not be horizontal, nor 
ab vertical, neither the height nor the breadth 



Fig. 38. 





Fig. 39. 



Fig. 39, a. 



CONSTRUCTIVE DRAWING. 45 

could be measured on these lines, but on lines that 
are horizontal and vertical, as ef and a'eb'. 

From what precedes, it will be seen that what- 
ever the form or position of the object to be de- 
scribed by Constructive Drawing, each part must be 
first measured; that is, the length and the breadth 
must be measured horizontally and perpendicular to 
each other; and the height must be measured verti- 
cally, perpendicular to both the length and the 
breadth. 

For the exact description of these dimensions, two 
different drawings are necessary. 

One of these drawings is always taken horizon- 
tally, parallel to the length and breadth, and is 
called the horizontal projection or plan. 

The other of these drawings is always taken ver- 
tically, parallel to the height, and is called the 
vertical projection or elevation. 

As it seldom happens that the object to be de- 
scribed is of such dimensions that it can be con- 
veniently drawn in its actual size, it becomes 
necessary to first understand what is meant by Scale 
Drawing. 



CHAPTER I.— Scale Drawing. 

Scale Drawings are of three kinds : Full Size, 
Eeduced, and Enlarged. 

In a Full Size scale drawing, every line in the 
object is described at its full length. 

In a Reduced scale drawing, every line in the 



46 MECHANICAL DRAWING. 

object is described a certain, constant number of 
times smaller than it is. 

In an Enlarged scale drawing, every line in the 
object is described a certain, constant number of 
times larger than it is. 



Section I. — Full Size Drawing. 

Prob. 1. — Represent a horizontal line 5 in. long. 
Fall size. 

Prob. 2. — Represent two parallel vertical lines 3f 
in. and 2J in. in length and J in. apart. Full size. 

Prob. 3. — Draw a circle If in. diameter. Full 
size. 

Prob. 4. — Draw a 2\ in. square. Full size. 

Prob. 5. — Draw a If in. square, diagonal vertical. 
Foil size. 

Prob. 6. — Draw a parallelogram 2 in.X } in., longer 
sides 30° with horizon to the right. Full size. 



Section II — Reduced Scale Drawing. 

Prob. 1. — Represent a line 8 ft. long. Reduced; 
scale, \ in. to the foot. 

Solution.— The statement, "Scale \ in. to the 

foot," signifies that 

B 1 ft. is the unit of 

measure, and J in. 
is the unit of representation. As there are 8 units 
of measure there must be 8 units of representation. 
Hence, AB represents 8 ft. because it is f in. in 



Scale, y± in. to the ft. 



CONSTR UCTIVE DBA WING. 47 

length (eight units of representation), and the scale 
is i in. to the foot. 

Prob. 2.— Draw a line that will represent 25 ft. 
Reduced; scale, i in. to the foot. 

Prob. 3. — Represent 100 ft. in length. Reduced; 
scale, j 1 ^ in. to the foot. 

Prob. 4. — Represent 5 ft. in length. Reduced; 
scale, 2 ft. to the in. 

Prob. 5.— Represent one side of the school-room 
door. Reduced; scale, J in. to the foot. 

Rule. — Multiply the number of units of measure 
in the line to be represented by the unit of repre- 
sentation, and the product will be the number of 
units of representation in the required line. 



Section III.— Enlarged Scale Drawing. 

Prob. 1. — Represent f in. in length. Enlarged; 
scale, f in. to J in. 

Solution.— In this problem, the unit of measure 
is J in. and the 
unit of representa- A p , ■ — — — B 

r Scale, % in. to % in. 

tion is fin. Hence, 

to represent f in. in length would require a line 
3X| in. =■£ = 2 in. AB, which is 2 in. long, rep- 
resents f in. when the scale is § in. to the J in. 

Prob. 2. — Represent f in. in length. Enlarged; 
scale, | in. to the i in. 



48 MECHANICAL DRAWING. 

JProb. 3. — Represent f in. in length. Enlarged; 
scale, J in. to J in. 

Prob. 4. — Represent Jg in. in length. Enlarged; 
scale, 32 in. to 1 in. 

Prob. 5. — Represent -^j- in. in length. Enlarged ; 
scale, J in. to T fa in. 

Prob. 6. — Represent a circle -f^ in. in diameter. 
Enlarged; scale, f in. to ■£% in. 

Prob. 7. — Represent an equilateral triangle ; side, 
- 9 7 in. in length. Enlarged; scale, J- in. to -fa in. 

Prob. 8. — Represent a regular octagon ^ in. in 
diameter. Enlarged; scale, 1 in. to -fa in. 

Rule. — Multiply the number of units of measure 
by the unit of representation, and the product will 
be the length of the required line in the drawing. 



CHAPTER II.-Plans. 

A Plan is a drawing which describes the length 
and breadth of an object on a surface that is con- 
sidered as always lying horizontal. 

Plans are of two kinds : parallel plans and an- 
gular plans. 

A Parallel plan is one that describes only lines 
and surfaces that are parallel to itself. 

An Angular plan is one in which the horizontal 
dimensions of lines or surfaces that are not parallel 
to itself are described. 



o 

Fig. 40. 



CONSTR UCTI VE DRA WING. 49 



Section I. — Parallel Plans. 

Prob. 1. — Draw a plan of a rectangular block that 
is 10 ft. long and 8 ft. wide; scale, y 1 ^ in. to the foot. 

Solution. — The block is rectangular; therefore, the 
plan will be rectangular. It is 10 ft. long 
and 8 ft. wide, and is to be represented on 
a scale y 1 ^ in. to the foot ; therefore, the 
plan will be a rectangle yf in. long and 
y^- in. wide. Draw a rectangle, ABCD, * 
(Fig. 40), | in. long and \ in. wide, 
and it will be the required plan of the block. 

Prob. 2. — Draw the plan of a rectangular block 
f< ^ 13 ft. long and 11 ft. wide; scale, J in. to the 
foot. 

Prob. 3.— Draw the plan of a triangular 
prism standing on end (Fig. 41), the sides of 
which are 2 ft., 2J ft., and 3 ft. wide ; scale, 
Fiff - 41 ' f in. to the foot/ 

Prob. 4. — Draw the plan of a circular 
cone. Base of cone, 19 ft.; scale, y^- in. 
to the foot. (Fig. 42). 

Prob. 5.— Draw the plan of a rectangu- 
lar block yL- in. wide and I in. long; 
scale, 1 in. to | in. rig ' 42 ' 

Prob. 6.— Draw a plan of a sphere (ball) 8000 
miles in diameter ; scale, J in. to 
the 1000 miles. 

Prob. 7. — Draw the plan of two 
rectangular blocks, 3 ft. long and 
Fig. 43. i fk wide, arranged as shown in 

the cut (Fig. 43). Scale, i in. to the foot. 



^K 





50 



MECHANICAL DRAWING. 



Prob. 8. — Draw the plan of a flight of six steps, 
each step 4 ft. long and 10J in. wide; scale, J in. to 

the foot. 
Zl Prob. 9.— Draw the 

plan of a flight of 6 
steps, with square 
landing, making a 
quarter turn midway 
of the flight. Steps, 
3 ft. X 101 in.; land- 




Fig. 44. 



™g> 



3 ft. 



square; 




Fig. 45 



Fig. 46. 



Fig. 47. 



scale, £ in. to the foot. (Fig. 44). 
Prob. 10.— Draw the plan 
of a flight of 9 steps, with 
two square landings, making 
two quarter turns, 3 steps be- 
tween the turns. Steps, 4 ft. 
X 1 ft. ; landings, 4 ft. square ; 
scale, i in. to the foot. 

Prob. 11.— Draw a plan of 
6 steps (Fig. 45), making a 
half turn in the middle by a land- 
ing 4 ft. X 8 ft. Steps, 1 ft. X 4 ft. ; 
scale, i in. to the foot. 

Prob. 12. — Draw the plan of a 

cubical block (Fig. 46), 4 ft. square, 

with a hole 18 in. square, extending 

through its center from top to bottom. 

Scale, % in. to the foot. 

Def.— A cube is a solid having six 
square faces. 

Prob. 13.— Draw the plan of a cylinder 
(Fig. 47), 24 in. diameter, 16 in, long, 



CONSTRUCTIVE DRAWING. 



51 



with a round hole 13 in. in diameter, extending 
lengthwise through its center; scale, ^ in. to the 
inch. 

Proh. 14. — A grindstone lying on the ground is 5 
ft. in diameter, with a hole 6 in. square in the center. 
Draw the plan; scale, J in. to the foot. 

Prob. 15. — Draw the plan of a cylinder tying on 
its side. Diameter, -g-% in.; length, T 6 g- in. ; scale, 16 
in. to the inch. 

Prob. 16. — Draw the plan of a flight of stairs con- 
sisting of 2 flyers and 

3 winders, making a 
quarter turn. Treads 
of flyers, 4 ft. X 1 ft. ; 
radius of inner curve 
of winders, 1 ft.; ra- 
dius of outer curve of 
of winders, 5 ft. ; scale, 
J in. to the ft. (Fig. 48). 

Prob. 17. — Draw the plan of a flight of 6 steps, 

4 flyers and 2 winders, making a quarter turn. 
Treads of flyers, 3 ft. X 10 J in.; outer radius of 

winders, 5 ft. 6 in.; inner ra- 
dius, 18 in.; scale, i in. to the 
foot. 

Prob. 18.— Draw the plan of 
a flight of 6 steps (Fig. 49), 
3 flyers, and 3 winders, mak- 
ing a quarter turn to the left 
hand. Treads of flyers, 10J 
in. X 3 ft. 6 in. Radius of 
outer circle of winders, 4 ft. 




Fig. 48. 



Lee 




Fig. 49. 



6 in.; scale, i in. to the foot. 

M. D.-5. 



52 MECHANICAL DRAWING. 

Prob. 19.— Draw the plan of a flight of 10 steps, 
4 winders, making a half turn to the right hand, 
and 3 flyers at either end of the flight. Treads of 
flyers, 3 ft. X 1 ft. Radius of inner curve of wind- 
ers, 9 in.; scale, J in. to the foot. 

Prob. 20. — Draw the plan of a flight of 12 steps, 
making two quarter turns to the left-hand; 3 wind- 
ers in each turn, with 2 flyers between them, and 2 
flyers at either end of the flight. Treads of flyers, 

3 ft. X 10J in. Radius of outer curve of winders, 

4 ft.; scale, f in. to the foot. 

Prob. 21. — Draw the plan of a flight of 12 steps, 
all winders, making a whole turn. Length of 
tread, 2 ft., 10| in.; outer radius of winders, 4 ft.; 
scale, i in. to the foot. 

Section II — Plans of the same object at different levels. 

When the horizontal dimensions of any object 
are the same at all heights, then the length and 
breadth of the object may be described in one plan. 

When the horizontal dimensions of any object 
vary at different heights, as in houses of more than 
one story, ships of more than one deck, etc., then 
the length and breadth of the object can be de- 
scribed only by as many different plans as there are 
different levels at which these dimensions change. 

The different plans required to describe all the 
horizontal dimensions of any object have each a 
name appropriate to the part of the object de- 
scribed. 

In buildings, we may have a ground plan, a base- 
ment plan, a story plan, and a roof plan. 



CONSTR UCTI YE DRA WING. 53 

A Ground Plan describes the length and breadth 
of the ground covered. 

A Basement Plan describes the length and 
breadth of the cellar. 

A Story Plan shows the length and breadth of 
every part of a building at a given story. 

A Roof Plan describes the length and breadth 
of every part of the roof of the object. 

Prob. 1. — Draw the ground plan of the school- 
house; scale, J in. to the foot. 

Prob. 2. — Draw the plan of the school-house 
grounds, and the ground plans of the buildings on 
it; scale, 64 ft. to the inch. 

Prob. 3. — Draw the ground plan of your school 
desk; scale, 1 in. to the foot. 

Note. — The ground plan of a portable object is the plan of 
that part that touches the ground or floor. 

Prob. 4.— Draw the ground plan of your teacher's 
desk ; scale, \ in. to the foot. 

Prob. 5. — Draw the ground plan and seat plan of a 
common wood-seat chair; scale, 1J in. to the foot. 

Prob. 6. — Draw the first-floor plan of your school- 
house ; scale, -^ in. to the foot. 

Prob. 7. — Draw the ground plan and 
the roof plan of two towers (Fig. 50), 
34 ft. outside diameters, 48 ft. high, 
38 ft. between centers, and connected 
by a covered passage-way 12 ft. wide 
and 12 ft. high, outside measurement; 
walls of towers, 24 in. thick, and walls Fig. 50. 

of passage, 12 in. thick; scale, -^ in. to the foot. 




54 



MECHANICAL DRAWING. 



Fig. 51. 



Prob. 8. — Draw the ground plan of the doorway 
of your school-room, with the door half open; scale, 
J in. to the foot. 

Prob. 9. — Draw the ground plan of 
a house (Fig. 51), 14 ft. X 30 ft., with 
10 ft. projection, 12 ft. long, in the 
middle of longer side; walls, 9 in. 
thick ; scale, Jin. to the foot. 
Prob. 10. — Draw the ground plans of two build- 
ings (Fig. 52), 48 ft. X 60 ft., with a 
hexagonal tower in the middle of 
one of the longer sides, and project- 
ing one half beyond the wall. One 
of these buildings is of brick, with a 
foundation Avail 24 in. thick; and one 
is a frame building, with a founda- 
tion wall 16 in. thick; scale, \ in. to 
the foot. 

Prob. 11.— A house 16 ft. X 20 ft. has a rectangu- 
lar projection at one corner (Fig. 
53). This projection has 2 walls, 5 
ft. apart and 4 ft. long, making 
angles of 135° with the end and side 
walls of the house. Draw the 
ground plan; walls, 15 in. thick; 




Fig. 52. 




Fig. 53. 



scale, \ in. to the foot. 




Prob. 12.— A house is 16 ft.X 36 ft., 
and has a bay window in the middle 
of one of the longer sides. This 
window, which has three equal sides, 
is 10 ft. wide and projects 3 ft. (Fig. 
54.) All the walls are 1 ft. thick. Fig * 54- 

Draw a ground plan ; scale, \ in. to the foot. 



CONSTRUCTIVE DRAWING. 



55 




Fig. 55. 



Prob. 13.— Draw the ground plan of a house 18 
ft. X 28 ft., extreme dimensions. 
One end is divided into 3 equal 
spaces, making 120° angles with 
each other and with the side walls 
of the house; scale, J in. to the 
foot. (Fig. 55.) 

Prob. 14.— A house 24 ft. square 

has a projection 12 ft. square at one end of one 

side, and in the angle formed by 

the house and the projection is a 

porch 12 ft. sq. (Fig. 56). Walls 

of house, 15 in. thick, and the 

outermost corner of the porch is 

supported by a pier 12 in. square. 

Draw ground plan of house and 

Fig - 56 - porch; scale, J in. to the foot. 

Prob. 15. — Draw the ground plan 

of your own house, or of any 

house that you may design. 

Prob. 16. — Draw the ground plan 
of an arch, 12 ft. span, standing 
on two piers, 3 ft. square ; scale, 
\ in. to the foot. 





Section III. — Angular Plans. 

Prob. 1. — Draw the plan of a gable roof 21 ft. 
long and 19 ft. span ; scale, -^ in. to the foot. 



Solution. — The length of a gable roof is measured 
on the eaves, and, unless otherwise specified, the 



56 



MECHANICAL DRAWING. 



eaves are horizontal, opposite, parallel, and equal, 
and the span is the perpendicular distance hetween 

them (Fig. 58). As the roof 
described in this problem is 
21 ft, long and 19 ft. span, 
and as the scale is ^ in. to 
the foot, we draw the rect- 
angle ABCD (Fig. 59), |i in. 
X 44 in., and as the ridge is 




Fig. 58. 



i o 

32 

equally distant from the eaves, 
we draw EF parallel to the 
lines representing the eaves and midway A B 

between them. ABFCDE is the re- 
quired plan of the roof, because it ex- 
actly describes the length and breadth 
of every part of the roof measured 
horizontally. 



Fig. 59. 



Prob. 2.— Draw the plan of a gable roof 18 ft. 
long and 14 ft. span ; scale, J in. to the foot. 

Prob. 3. — A gable-roofed house 
(Fig. 60), 30 ft. long and 18 it. wide, 
has a wing in the middle of one of 
its sides, 16 ft. long and 6 ft. projec- 
tion, covered by an extension of 
the roof of the main house; roof 
projection beyond the wall of the house 
on all sides, 1 ft. Draw the plan of the 
roof; scale, i in. to the foot. 

Prob. 4. — Draw the plan of a rectan- 
gular pyramid with the apex vertically 
over one corner of the base (Fig. 61). 
Base, 9 ft. X 6 ft. sq.; scale, J in. to the ft. 



Fig. 60. 




Fig. 61. 



CONSTRUCTIVE DRAWING. 



57 




Fig. 62. 



Prob. 5.— Draw the plan of a frustum of a rect- 
angular pyramid. Base, 11 ft. X 
13 ft.; and deck, 6 ft. X 6 ft. 6 in. 
Hips, equal in length; scale, J in. 
to the foot. (Fig. 62.) 

Prob. 6. — Draw the plan of a 

hooded-gable roof. Ridge, 19 ft.; 
eaves, 30 ft. Span of gable at the 
top, 8 ft. Span at the bottom, 18 
ft. Scale, ^ in. to the foot. (Fig. 
63.) 

Prob. 7. — Draw a plan of the roof 
of a gable-roofed house with wing. 
House, 20 ft, X 24 ft. Wing, 8 ft. 
projection, 16 ft, in length, 
and in the middle of the 20 
ft. side of the house. Eaves 
and the gable of the roof 
project 18 in.; scale, -^ in. to 
the foot, (Fig. 64.) 

Prob. 8. — Draw the plan of 
the roof of a gable-roofed 

house and wins- 




Fig. 63. 




Fig. 64. 




Fig. 65. 



The ridge of 
the roof of the wing, even with 
the eaves of the main house. 
Main roof, 16 ft, span and 24 ft. 
long. Wing roof, 12 ft, square, 
and in the middle of the longer 
side of the house ; scale, -^ in. 
to the foot. (Fig. 65.) 



Def. — A rafter is a roof timber extending from 



the eaves to the ridge. 



58 



MECHANICAL DRA WING. 




Prob. 9. — Draw the plan of a gable-roofed house 
with curved rafters. Eaves, 16 ft. 
long; span, 14 ft.; scale, -^ in. to 
the foot, (Fig. 66.) 

Prob. 10. — Draw the roof plan 

of a gable-roofed building with 

smaller gable-roofed extension at 

one end. Main house, 20 ft. wide 

long. Extension, 12 ft, 



Fig. 66. 



and 24 ft 

square, 6 ft. from right-hand side of 
end. Roof projection, 18 in. on all 
sides; scale, J in. to the ft. (Fig. 67.) 
Prob. 11. — Draw the top plan of 
your school desk; scale, 1 in. to the 
foot. 

Section IV. — Condensed Plans. 




Fig. 67. 



Whenever it can be done without rendering the 
description any less clear and unmistakable, it is the 
custom among draughtsmen to describe all the hori- 
zontal dimensions belonging to the same object, in 
one drawing, called a condensed plan. 

Of the different plans which constitute a con- 
densed plan, one is called the Principal Plan. 

The Principal Plan may be the plan of the 
object at any level, and is always drawn in a con- 
tinuous line. 

Plans of levels above the principal plan may be 
drawn in lines made up of dashes and hyphens; as, 
or or , etc. 

Plans of levels below the principal plan may be 
drawn in lines made up of dashes and dots; as, 
_ . _ . _ . or —..._.. . or _.._.. , etc. 



CONSTRUCTIVE DRAWING. 



59 



Prob. 1. — Draw a con- 
densed ground plan of a 
box 2 ft. 9 in. sq., made of 
planks 3 in. thick, and 
having a lid projecting 6 
in. all around ; scale, \ in. 

to the foot. 





Fig. 68. 

(Fig. 68.) 



Scale l 4 in.to the ft. 



1 1 

j! 


B 


!i 


M__. 


.-_- 


-,J\ 



Fig. 69. 



Solution. — In a condensed 
ground plan, the ground plan of 
the box is a continuous line, 
and the plans of the lid and in- 
side are drawn in dotted lines, A, 
Fig. 69. 

If the problem called for a con- 
densed lid plan, it would be drawn 
as shown in i>, Fig. 69. 



Prob. 2. — Draw a condensed 

middle plan of a common spool ; 

scale, 2 in. to the in. (Fig. 70.) 

Prob. 3. — Draw 

a condensed lid 

plan of a box 4 

ft. wide and 6 ft. 

3 in. long, made 

of 3 in. plank, 

with lid project- 
Fig. 70. . « • 11 
mg o m. alL 

round, and having a row 

(dentils) 3 in. square and 6 in. apart on all sides 

immediately below the lid; scale, f in. to the 

foot. (Fig. 71.) 





Fig. 71. 

of rectangular 



locks 



60 



MECHANICAL DRAWING. 




Prob. 4. — Draw a condensed ground plan of a 
cubical block 16 ft. square, with a 
hole 9 ft. square extending through 
its center from top to bottom, and 
an opening 3 ft. wide and 6 ft. 
high extending through one of its 
Fig. 72. sides at the middle of the lower 

edge ; scale, T \- in. to the foot. (Fig. 72.) 

Prob. 5. — Draw a condensed 
ground plan of a frame- work 4 
ft. wide, 7 ft. long, and 3 ft. high, 
made of timbers 15 in. square; 
scale, J in. to the foot. (Fig. 73.) 
Prob. 6. — Draw a condensed 
middle plan of a block 10 ft. 
square and 5 ft. high, with panels on each of the 
four vertical sides, 5 ft. long, 2 
ft. 6 in. high, and 6 in. deep, bev- 
eled on the edge of the panel 
opening at an angle of 45°; scale, 
| in. to the foot (Fig. 74.) 




Fig. 73. 











\ 










j2 


\ 









K 



Fig. 74. 



Def. — A bevel is a slanting edge. 



Prob. 7. — Draw a condensed ground 
plan of a bobbin. Shank, 1J in. in diam- 
eter, 6J in. long between the flanges, 
and a hole § in. diameter lengthwise 
through the center; ends, 4| in. diame- 
ter, | in. thick, and square edge; scale, 
J in. to the inch. (Fig. 75.) 

Prob. 8. — Draw a floor plan of the 
school-room; scale, I in. to the foot. 




Fig. 75. 



CONSTRUCTIVE DRAWING. 



61 



Note. — There are cases in actual practice in which certain 
facts of construction are represented in 
a way peculiar to themselves, without 
regard to any rule of constructive draw- 
ing. This is true of windows ; for, in a 
strictly floor plan, the windows would 
not be represented at all, because they 
do not come to the floor. For conven- 
ience sake, draughtsmen have adopted 
the plan of representing all windows as 
shown in W. and all doors as shown in 




A Fig. 76. 





f c Jk 

/ 

Span_^ 





Fig. 77. 



rn 



o 



Fig. 76. 

Prob. 9. — Draw the plan of a wall 3 
ft. thick, 25 ft. long and 15 ft, high, 
pierced by two equal openings 8 ft, 
wide, having a semicircular arch at the 
top, and 8 ft. high to the springing of 
the arch ; scale, \ in. to the foot, 
(Figs. 77 and 78.) 



Fig. 78. 



Note. — Whenever it is not expressly stated, 
it is expected that the draughtsman will exer- 
cise his judgment in condensing his plans. 
His whole aim should constantly be to state 
his facts as briefly, compactly, and clearly as 
possible, without regard to time, labor, or the 
number of drawings required. 



CHAPTER III.— Elevations. 

An Elevation is a drawing made on a flat surface 
that is regarded as always standing vertical — per- 
pendicular to the plan. 

Heights or Yertical Dimensions are always de- 
scribed by the elevation because the elevation itself 
is always vertical. 



62 



MECHANICAL DRAWING. 



The lines and surfaces that may he described by 
an elevation are not always vertical, and therefore 
they are not always parallel to it. Hence, there are 
two kinds of elevations : parallel elevations and 
angular elevations. 

A Parallel Elevation describes only lines and 
surfaces that are parallel to itself. 

An Angular Elevation describes lines and sur- 
faces that are not parallel to itself. 

Both parallel and angular elevations may describe 
interior or exterior heights, and it not infrequently 
happens that to completely describe the heights of 
all parts of an object, many different elevations of 
both the exterior and interior are required. 

Each elevation must have a title inscribed upon 
its face indicating the side and part of the object 
described. 

To describe stationary objects, the elevations may 
be named according to the points of the compass, 
or indicated as front, rear, end, or side elevations. 

Elevations describing portable objects must have 
appropriate names devised by the draughtsman. 

Section I. — Exterior Parallel Elevations. 




Fig. 79. 



Prob. 1.— A house, 13 ft, X 16 ft., 
with walls 6 in. thick and 11 ft, high, 
including foundation 1 ft. 6 in., and 
with a gable roof, 6 ft. pitch (vertical 
height), projecting 9 in. beyond the 
walls on all sides, stands with its longer 
sides north and south. There is a 
door, 2 ft. 6 in. X 6 ft, 6 in., 2 ft, from 



CONSTRUCTIVE DRAWING. 



63 



the westerly end in the south wall. The windows 
are all 2 ft. 3 in. X 5 ft., 2 ft. 3 in. above the top of 
the foundation. In the south wall is one window 
midway between the door and the easterly end. In 
the east wall are two windows, 3 ft. apart, and 
equally distant from the ends. In the north wall 
there is one window, in the middle. In the west 
Avail are two windows, 2 ft. 3 in. from the ends. 
Draw the plan and elevations of each side; scale, 
JL i n . to the foot. (Fig. 79.) 

Solution. — Regarding the top of the drawing as 
north, draw a condensed 
plan, A, (Fig. 80), ft in. 
wide by -Jf in. long, the 
longer side extending 
north and south, and lo- 
cate in it the width and 
position of the door and 
windows. From each side 
of the plan draw perpen- 
dicular leading (clotted) 
lines. Perpendicular to 
each set of leading lines, 
and at a distance from the plan greater than the 
entire height of the object to be represented, draw 
the base-line W. From this base-line lay off 
towards the plan on each leading line the height of 
the object at that point, and at these points draw 
the necessary connecting lines to complete the 
several elevations, B, B, _B, and B. 

Prob. 2. — Draw the plan and parallel elevations 
of a cubical block 4 ft. square ; scale, \ in. to the ft. 




1 
( 

> 


H 



Scale, ^ in. to the foot. 
Fig. 80. 



64 



MECHANICAL DRAWING. 



Prob. 3. — Draw the plan and parallel elevations of 
the end and side of a rectangular block 6 ft. 6 in. 
long, 5 ft. wide, and 4 ft. high ; scale, J in. to the ft. 
Prob. 4. — Draw the plan and parallel end and side 
exterior elevations of a rectangular 
block 3 ft. 7 J in. long, 4 ft. 4 J 
in. wide, and 4 ft. high, with two 
holes 19J in. square, one 3 ft. 7| in. 
long, and the other 4 ft. long, per- 
pendicular to each other and to 
the faces of the block, and passing 
• ; scale, J in. to the foot. (Fig. 81.) 



Fig. 81. 




through its centei 

Prob. 5. — Draw the plan and 
the parallel elevation of two 
adjacent sides of two timbers, 
laid at right angles each across 
the middle of the other. The 
timbers are each 18 in. wide 
and 6 in. thick; one of them 
the other is 12 ft. lon^; scale, 
(Fig. 82.) 

Prob. G. — Draw the plan and the parallel end and 
side elevations of a box 13J 



Fig. 82. 

is 6 ft. long, and 
r in. to the foot. 




ft. 



lOUOf, Or 



* ft. wide, and 6 ft. 



high, with a top 3 in. thick, 
projecting 6J in., with 2 den- 
tils 3 in. square at each cor- 
ner, immediately below the 
top; scale, I in. to the foot. 
(Fig. 83.) 
Prob. 7. — Draw the plan and the parallel end and 

side elevations of the teacher's platform; scale, i 

in. to the foot. 



Fig. 83. 



CONSTRUCTIVE DRAWING. 



65 




Fig. 84. 



Prob. 8. — Draw plan and parallel end and rear ele- 
vations of your school desk ; scale, 1 in. to the foot. 

Prob. 9. — Draw the plan and the end and front 
elevations of two steps. Treads, 1 
ft. X 3 ft., and risers, 6 in.; scale, \ 
in. to the foot. (Fig. 84.) 

Prob. 10. — Draw the plan and the 
elevation of a flight of 6 steps, mak- 
ing a quarter turn in the middle, to the right hand, 
by a square landing. Treads, 1 ft. X 4 ft., risers, 7J 
in.; landing, 4 ft. square; scale, J in. to the foot. 

Prob. 11.— Draw the plan and the parallel end and 
side elevations of a wall 3 ft. thick, 15 ft. high, and 
25 ft. long, pierced by two circular arched openings, 
8 ft. wide, and 8 ft. high to the springing of the 
arches; scale, J in. to the foot. 

Prob. 12. — Draw the plan and the parallel front 
and side exterior elevations of a flat-roofed house 16 
ft. X 24 ft,; height of wall, 14 ft., including the 
foundation of 2 ft.; roof-cornice extending across the 
front, 1 ft. projection, supported by 9 brackets, 10J X 
12 in., and equally distant apart; the chimney, 24 in.X 
18 in., and extending 4J ft, above the roof, is in the 
rear wall, 1 ft. from the right-hand side of window ; 
there is a window in the middle of both the longer 
sides, and one in the middle of one end; a door is 
in the middle of the front. The windows are 2 ft. 
6 in. X 6 ft,, and 2 ft. 6 in. from the floor ; the door 
is 3 ft. wide and 7 ft, high; scale, \ in. to the 
foot. 

Prob. 13. — Assume the dimensions and draw the 
plan and the elevation of a gable-roofed house with 
front and side porches. 



66 



MECHANICAL DRAWING. 



Section II. — Interior Parallel Elevations. 



Prob. 1. — A school-room is 32 ft. wide, 36 ft. long, 
and 12 ft. high. In the front Avail there are two 
doors 3 ft. Avide and 7 ft. high, 2 ft. 6 in. from the 
end of the wall ; in each of the side walls are 3 
windows, 3 ft. wide X 6 ft. 6 in. high, 3J ft. from 
the floor, and 6 ft. 6 in. apart. There is a black- 
hoard 4 ft. wide and 2 ft. 6 in. from the floor on 
the front wall between the doors, and another ex- 
tending entirely across the rear wall. Draw the 
plan and the parallel interior elevation of each 
wall; scale, ■£% in. to the foot. 



Solution.- 



-Draw the plan, 



8 2 

"G4 



111. 



wide and |-| in. 



p: 



,-A 



Cd 



long, showing the widths and 
positions of the doors and win- 
dows, A (Fig. 85). Draw from 
each side of this plan leading 
lines, and perpendicular to these 
draw the base-lines b'b r . From 
these lines lay oft* on the lead- 
ing lines, away from the plan 
(see note), the heights of the 
different parts of the wall above the floor. Draw 
lines joining these points, and they will enclose the 
elevations of the several walls P, P, P, and P. 



i, Hllllllllih'i'lMlil 

6 



Fig. 85. 



Note. — Compare this solution with that on page 63, and it 
will be seen that the only difference between an interior and 
an exterior elevation is that in an interior elevation the base- 
line is the line of the elevation that is nearest to the plan, 
and in an exterior elevation the base-line is the line of the 
elevation that is farthest from the plan. 



CONSTRICTIVE DRAWING. 67 

Prob. 2. — Draw the plan and the interior elevation 
of the front wall of the school-room and the teach- 
er's platform; scale, J in. to the foot. 

Prob. 3. — Draw the plan and the interior elevation 
of the rear wall of the school-room and the pupils' 
desks; scale, J in. to the foot. 

Prob. 4.— A room, 25 ft, X 21 ft,, and 10 ft. high, 
has five windows, 30 in. wide, and 6 ft. 6 in. high, 2 
ft. from the floor; two doors, 36 in. wide and 7 ft. 6 
in. high ; and a chimney, 2 ft. breast (width across 
the front) and 1 ft. jamb (projection from the wall). 
The length of the room stands north and south. 
The chimney is in the middle of the east wall, with 
a window on either side and 3 ft. from it; one door 
is opposite the chimney, and the other door is in the 
middle of the south wall, with a window on either 
side, and 2 ft. 6 in. from it; and one Avindow is in 
the middle of the north wall. Draw the plan and 
the interior elevation of each wall; scale, J- in. to 
the foot. 

Prob. 5. — Draw the floor plan and the interior 
parallel elevations of each side of your school desk ; 
scale, 1 in. to the foot. 

Prob. 6. — Draw the plan and four interior parallel 
elevations of the walls of any hall in the school 
building; scale, J in. to the foot. 

Section III. — Condensed Elevations. 

As in making plans time and space may be 
economized, when the object represented is not too 
complicated, by drawing two or more plans within 
the same outline, so in making elevations when 

M. D.— 6. 



68 



MECHANICAL DRAWING. 



there are not too many parts to be represented, ex- 
terior and interior elevations may be made in the 
same drawing. 

Prob. 1. — Draw a condensed plan and elevation 
of a box 8 in. square and 7 in. high, 
ZZZ outside measurement, and made of boards 
1 in. thick. The sides are mitered, the 
bottom is let in, and the top is laid on ; 

Fig. 86. scale, J in. to the inch. 

Solution. — Draw the plan ABCD, § in. square = 
1 in. As the sides extend to . ~ 

A 

the ground, the plan of the in- 
side of the box will be drawn 
in full lines, EFGH, and \ in. 
from the sides of ABCD, be- 
cause the sides are 1 in. thick. 
The joints between the sides at 
the corners will be represented 
by diagonal lines, AI, FB, GO, 
and HD, joining the plan of 
the outside and inside of the 
box. Draw the elevation dcih, 
| in. high. Draw a full line 
jk, J in. below the top, to rep- 
resent the bottom of the lid. Flg ' 87 ' 
Draw Im and no to represent the interior elevation 
of the side, and pt to represent the interior eleva- 
tion of the bottom, showing that it is let in. E is 
the elevation and P is the plan, and together they 
completely and clearly describe the dimensions and 
construction of the box required in the problem. 
(Fig. 87.) 



dL 




E 




CONSTR VCTIVE DRA WING. 69 

Prob. 2. — Draw the plan and as many elevations 
as may be necessary to com- 
pletely and clearly describe a 
box 31 in. square and 17 in. 
high, made of planks 3 in. 
thick, and having no side let 
in at the ends; scale, J in. to 
the inch. (Fig. 88.) 

Prob. 3.— A brick wall 16 in. Fig ' 88 ' 

thick and 12 ft. high encloses a space 24 ft. X 30 ft. 
The shorter walls are blank ; there is a door 3 ft. X 
7 ft. 6 in., 1 ft. from the ground, reached by two 
steps, 3 ft. X 1 ft. tread, and 6 in. rise; and there 
are two windows 2 ft. 10 J in. wide, 6 ft. 6 in. long, 
and 2 ft. from the floor, dividing the opposite wall 
into three equal spaces. Make one plan and eleva- 
tion which completely describes the construction ; 
scale, J in. to the foot. 

Prob. 4. — Draw one plan and one elevation that 
completely describes the dimensions and construc- 
tion of a tool box or a work box ; scale, \ in. to the 
foot. 

Section IV. — Angular Elevations. 

An Angular Elevation describes lines and sur- 
faces that are not parallel to itself. 

Prob. 1. — Draw an angular elevation of two sides 
of a cube 4 ft. square ; scale, J in. to the foot. 

Solution. — Draw a J in. square to represent the 
plan of a cube 4 ft. square. Draw AB, the base- 



70 



NEC HA NIC A L BRA WING. 



me of the elevation, at an angle with the sides 
1-2 and 2-3. From 1, 2, and 3 
draw leading lines perpendicular 
to the base-line AB. On these 
lines lay off, one inch from the 
base-line, l'l", 2'2", and 3'3" for 
the height. Draw 1"2"3", and 
1'1"2"3"3'2' is the outline of an 
angular elevation of two sides 
of a cube 4 ft. square. (Fig. 
89.) 




Fig. 89. 



Prob. 2. — Draw an end, a side, and an end and 
side elevation of a house 16 ft. wide, 23 ft. long, 
and 14 ft. high, including the foundation, which is 
2 ft. high. Gable roof, 8 ft. pitch and 1J ft. projec- 
tion ; bay window in middle of end, 10 ft. wide, 11 
ft. high, in- 
cluding foun- 
dation, 4 ft. 
projection, 3 
equal sides, 
and gable 
roof J pitch ; 
scale, -gL- in. 
to the foot. 










Solution. — 
D r a w the 
ground plan, 
abed, if in. 
wide and -|-| 
in. long, showing the angle and projection of the 
bay window, E. Draw the roof plan, ABCD, -|-| 




Fig. 90. 



CONSTR UCTIVE DRA WING. 71 

in. wide and |-| in. long in dotted line. Draw lead- 
ing lines from the plan, and construct the eleva- 
tions if, N, 0. (Fig. 90.) 

Prob. 3. — Draw a plan and elevation of a pyramid 
4 ft. square at the base and 6 ft. pitch; scale, % in. 
to the foot. 

Prob. 4. — Draw a plan and elevation of a hex- 
agonal prism 18 in. in diameter and 14 in. long; 
scale, Jg- in. to the inch. 

Prob. 5. — Draw a plan and elevation of a cone 3 
in. diameter and 1J in. pitch ; scale, 1 in. to the 
inch. 

Prob. 6. — Draw a plan and elevations of a sphere 
-^2 in. in diameter; scale, 1 in. to -^ in. 

Note. — The preceding problems require the description of 
objects so formed that a parallel elevation can not be made, 
and therefore the term angular elevation is omitted. 

Prob. 7. — Draw a plan and angular elevation of a 
cube 5 ft. 6 in. square; scale, \ in. to the foot. 

Prob. 8. — Draw a plan and angular front and side 
elevation of your school desk; scale, 1 in. to the 
foot, 

Prob. 9. — Draw a plan and a front elevation, a 
side elevation, and a front and side elevation of a 
house 21 ft. wide and 33 ft. long; 14 ft. high, in- 
cluding a foundation of 18 in.; Hat roof; bay win- 
dow, 10 ft. wide, 3 ft. 6 in. projection, 3 equal sides; 
scale, J- in. to the foot. 

Prob. 10. — Draw a plan and an interior angular 
elevation of a corner of the school -room, taking in 
12 ft. of the walls forming the corner; scale, \ in. 
to the foot. 



72 MECHANICAL DRAWING. 

Prob. 11. — A room is 21 ft. long, 15 ft. wide, and 
10 ft. high; it has seven windows 3 ft. wide, 6 ft. 
high, and 3 ft. from the floor; two doors 3 ft. wide, 
7 ft. high ; one chimney 5 ft. wide (breast), 18 in. pro- 
jection (jamb), — chimney in the middle of the 
shorter side ; three windows 3 ft. apart and 3 ft. 
from the end, in the longer wall to the right hand 
of chimney ; one window opposite chimney, and 
one door on either side of it and 3 ft. from it; one 
window in the middle of the longer wall to the left 
hand of chimney. Draw plan and angular eleva- 
tions of each corner, taking in the entire walls 
forming each ; scale, J in. to the foot. 

Prob. 12.— Draw a plan and angular elevation of 

a chimney 6 ft. breast and 
21 in. jamb, with mantle 
10| in. wide, and 3 in. 
thick, 5 ft. from floor, and 
fire-place opening 2 ft. 
square and 9 in. jambs; scale, J in. to the foot. 

Prob. 13. — Draw a plan and angular elevation of 

the school-house steps ; scale, J in. to the foot. 

Prob. 14.— Draw plan and elevation of common spool. 

Prob. 15. —Draw a plan and angular elevation of a 

flight of 4 steps, 12 in. by 3 ft. tread, and 7J in. riser, 

with a rail 2 in. thick and 4 in. wide, supported by 

one baluster at each step 1J in. from the front and 

end of the tread and 3 ft. high ; scale, J in. to the foot. 

Prob. 16. — Draw a plan and angular elevation of a 

cylinder lying on its side; diameter, 3 in.; length, 

4 J in.; scale, I in. to the inch. 

Prob. 17. — Draw a plan and angular elevation of 
a hexagonal pyramid 21 in. long and 27 in. in diam- 



^* 








<, 


"5t 






E 

CS 
•"5 


| 












B 




91. 




1 


Breast 


Mg. 


Breas 


t 



CONS TR UC Tl VE BRA WING. 



73 




li 



Fig. 92. 



eter, lying on its side, and its axis making an angle 
of 45° with the elevation; scale, -^ in. to the inch. 
Prob. 18. — Draw plan and an angular elevation of 
a barrel lying on its side ; diameter at the head, 18 
in.; and at the bilge, 21 in.; length, 33 in.; scale, -^ 
in. to the inch. 

Prob. 19.— Draw plan and angular elevation of a 
common spool lying down with its axis 
making an angle of 30° with the eleva- 
tion ; scale, 2 in. to the inch. 

Prob. 20.— Draw a plan and angular 
elevation of a spool with straight 
flanges (Fig. 92). Shank, 1 in. in diame- 
ter, 3 in. long inside the flanges; flanges, 
in. wide and 1 in. thick, and bore 

through the center f in. in di- 
ameter; scale, 1J in. to the inch. 
s Prob. 21. — Draw a plan and any 

angular elevation of a pulley 3 
ft. in diameter; rim, 6 in. wide, 
3 in. thick ; hub, 9 in. in diame- 
ter, 6 in. long, and bore 3 in. in 
Fig. 93. diameter; 4 spokes 1 in. X 5 in.; 

scale, 1 in. to the foot. (Fig. 93.) 
Prob. 22.— A piece of timber 13 
in. square and 10 ft. long rests 
one end on the ground and the 
other end is 5 ft. from the ground, 
in the angle formed by two 
walls, the foot of the timber being 
equally distant from the walls 
the angle. Draw plan 





forming 

and elevation ; scale, J in. to the 



Fig. 94. 

foot. (Fig. 94.; 



74 



MECHANICAL DRAWING. 



Prob. 23. — Draw a plan and front, side, and rear 
elevations of a common chair; scale, J in. to the ft. 



-JfC' 



Section V. — Two or more parallel elevations on the 
same base-line. 

Prob. 1. — Draw the plan and parallel elevations 
of two adjacent sides of a box 6 in. long, 3 in. 
wide, 4J in. high, J in. thick; scale, J in. to the 
inch. 

Solution. — Draw the plan, abed, and draw the 
elevation, a'a'b'b' . Swing the 
side cb about the corner b un- 
til c"b f is in line with ab. 
Now lead c" down to the base- 
line a'b, and b'b'c'c' is the ele- 
vation of the side CB, and 
we have parallel elevations 
of two adjacent sides of an 
object on the same base-line. 
(Fig. 95.) 
Prob. 2. — Draw the plan and elevations of two 
adjacent sides of a rectangular pyr- 
amid parallel to one side of the 
base. Base, 4 ft. X 7 ft., and pitch 
8 ft.; scale, -Jg- in. to the foot. 

Solution. — Draw the plan abode. 
Draw the end elevation, a'b'e'. 
Swing the side bee about the point 
b, into line with ab. Draw leaders 
from c" and c", and construct the 
elevation b'e'e'. (Fig. 96.) 



b 

Fig. 95 




CONSTRUCTIVE DRAWING. 



75 



Prob. 3. — Draw the plan and parallel elevations of 
two sides of a box 6 in. wide, 5 in. long, 3 in. 
deep, and 1 in. thick. Bottom let in, ends laid on, 
and lid projecting 1 in. all around; scale, J in. to 
the inch. 

Prob. 4. — Draw the plan and parallel elevations of 
two sides of a rectangular frame-work 10 ft. long, 
7 ft. wide, 9 ft. high, and made of timber 6 in. X 12 
in., with the wider side of timbers vertical; scale, f 
in. to the foot. 

Prob. 5. — Draw the plan and three elevations par- 
allel to one of the sides of a triangular pyramid. 
Bases of sides, 3, 5, and 7 in., and pitch 4 in.; scale, 
i in. to the inch. 

Prob. 6. — Draw a plan and parallel elevations of 
three sides of a 

house 19 ft. sq., 16 :^ ^ 

ft. high, including 
foundation, 2 ft. ; 
gable roof, 6 ft. 
pitch and 1 ft. pro- 
jection. Door in 
middle of one end 
4 ft. wide and 9 ft. 
high. Windows, 

2 ft. 9 in. X 8 ft,, Fig - 97.-First Method. 

2 ft. 3 in. from the floor, one in the middle of end 
opposite the door, one in the middle of the right- 
hand wall, and two dividing the remaining wall into 
three equal spaces on the outside; walls, 9 in. thick; 
scale, J-j in. to the foot. 

Solution. — First Method: Draw the plan and con- 

M. D —7. 




76 



MECHANICAL DRAWING. 



struct the elevations as in the preceding problems, 
and it will be seen that because the roof projects 
beyond the body of the house, the gable and the 
eaves of the front and side elevations lap over one 
another (see first method, Fig. 97). This tends to 
confound two different drawings, and in many cases 
might lead to confusion and misunderstanding. 
To avoid this, a method slightly different is adopted 
by draughtsmen : 




b' b' 

Fig. 98— Second Method. 



Second Method. — Draw the plan. Draw lines 
parallel to and at any convenient distance from the 
sides to be revolved, a"d n and b"c". Draw leaders 
connecting these lines with the sides of the plan. 
Revolve these lines as in the preceding method, and 
construct the elevations. By this method, each ele- 
vation is entirely separate and distinct from the 
others. (Fig. 98.) 

Prob. 7.— Draw the plan and elevations of three 



sides of a box 6 



in 



long, 



b in. 



wide, 4J in. high, 



and J in. thick. Mitered corners, bottom put on. 



CONSTRUCTIVE DRAWING. 77 

and lid projecting 1 in. on all sides ; scale, J in. to 
the foot. 

Prob. 8. — Three timbers, one 4 ft. long, one 6 ft. 
long, and one 8 ft. long, and all 1 ft. 
square, are framed together perpendic- 
ular to each other at the middle. Draw 
plan and two elevations ; scale, i in. to 
the foot. (Fig. 99.) 

Prob. 9. — Draw plan and elevations Fig * "• 
of three sides of your school desk; scale, J in. to 
the foot. 

Prob. 10. — Draw plan and elevations of four 
sides of your school-house ; scale, ^ or ^ in. to the 
foot. 




CHAPTER IV.— Points in Inclined Surfaces. 

Section I. — To find the elevation of a point in the edge 
of an inclined surface when the plan and the ele- 
vation of the surface and the plan of the point are 
given. 

Prob. 1.— The plan of an inclined surface is a 
rectangle 7 ft. X 8 ft. The elevations of the 8 ft. 
edges are horizontal, and the elevations of the 7 ft. 
edges have a pitch of 6 ft. In the right hand in- 
clined edge of this surface is a point, and the plan 
of this point is in the right hand shorter edge of 
the plan, 4 ft. from the plan of the lower edge. It 
is required to describe this point and the surface by 
a plan and elevation drawing; scale, ^ in. to the 
foot. 



78 



MECHANICAL DRAWING. 



Solution. 




Fig. 100. 
the surface it wi 



As the plan of the given inclined sur- 
face is a rectangle 7 ft. X 8 ft., and 
the scale is T a g in. to the foot, draw a 
rectangle, abed, T 7 g X T 8 g in. From 
this plan draw leaders, and construct 
an angular elevation, a'b'c'd', having 
a pitch of ^g- in., ed'. This fully de- 
scribes the given inclined surface. 
As the plan of the given point is in 
the right hand side of the plan of 
be in be. As it is 4 ft. from the 



lower horizontal edge, it will be at p, -^ in., from 
ah on be. It is required to find its elevation. As 
the plan of the point is on be, its elevation must be 
on the elevation of BC=b f c'. It must also be on a 
leader drawn from p to intersect b'c' in p f . The 
point P, in the edge of the inclined surface, is now 
completely described by its plan and elevation p and 
p'. (Fig. 100.) 

Prob. 2.— An inclined surface has a rectangular 



plan, 



41 
^2 



X 7| in.; the elevations of the shorter 



edges are horizontal, and the pitch of the other 
edges is 3J in. In the left-hand inclined edge is a 
point, and the plan of this point is 1\ in. from the 
plan of the upper edge of the surface. Draw a 
plan and elevation that will completely describe the 
given point and surface; scale, J in. to the inch. 

Pro!>. 3. — A wedge stands on 
its base, which is 9 ft. X 18 ft., 
and the pitch is 8 ft. In each 
of the edges of one of its sides 
is a point, a, b, c, and d. The Fig. 101. 

plan of a is on the plan of the ridge 4 ft. from the 




CONSTRUCTIVE DRAWING. 79 

right-hand end; the plan of b is on the plan of the 
right-hand edge, and 2J ft. from the plan of one 
of the longer edges of the base; the plan of the 
point c, in the longer side of the base, is 8 ft. from 
the left-hand end; the plan of the point d is in the 
plan of the left-hand edge and 4 ft. from the plan 
of the ridge. Draw plan and elevation ; scale, J in. 
to the foot. (Fig. 101.) 

Prob. 4. — A rectangular pyramid, base 12 in. X 9 
in., and pitch 18 in., stands on its base and has a 
point in each of the edges of one of its 
larger triangular sides, a, b, and c. 
(Fig. 102.) The plan of a is on the 
plan of the left-hand edge of the side, 
4 in. from the plan of the base; the 
plan of the point c is in the middle of 
the plan of one of the longer sides of 
the base; and the plan of 6 is in the plan of the 
right-hand edge 3f in. from the plan of the apex 
of the pyramid. Draw plan and elevation ; scale, 
j^- in. to the inch. 

Prob. 5. — Draw plan and elevation of an oblique 
triangular pyramid; sides of base, 3 ft.; pitch, 4 J 
ft.; and the plan of the apex on a line bisecting one 
of the angles of the base, and 5 ft. from it; scale, 1 
in. to the foot. 

Rule.— Draw the plan and elevation of the given 
inclined surface, and locate the given plan of the 
required point. From this plan of the point, draw 
a leader to intersect the corresponding edge of the 
elevation, and this point of intersection will be the 
required elevation of the given point. 




80 



MECHANICAL DRAWING. 



l 



Section II. — To find the elevation of a line that joins 
two edges of an inclined surface when the plan 
and elevation of the surface and the plan of the 
line are given. 

Prob. 1.— The plan of an inclined surface is a 
rectangle 25 ft. X 32 ft.; the elevation of the 32 ft. 
edges are horizontal, and the pitch of the other 
edges is 22 ft., and the plan of a line crossing this 
surface has one end in the left-hand edge of the 
plan of the surface 5 ft. from the plan of the lower 
horizontal edge, and the other end is in the plan of 
the upper edge of the surface, 6 ft. from the right- 
hand edge of the plan. It is required to draw the 
plan and elevation of the surface and the line; scale, 
in. to the foot. 
Solution. — Draw a rectangle, abed, -Jf in. Xff in., 
for the plan of the given surface. 
From this draw the angular eleva- 
tion, a'b'c'd', with the longer edges 
horizontal, and with a pitch of ff 
in. Locate the ends, p and q, of the 
plan of the given line pq, p on ad, 
■fy in. from dc, and q on ab> -fy in. 
from be. Join p and q, and pq is 
the plan of the given line. Find 
the elevations of p and q in p' and 
</', and join them, and p' (f is the ele- 
vation of the given line. 

Prob. 2. — One side of a gable roof is 16 ft. long, 
10 ft. span, and 8 ft. pitch (the plan a rectangle 16 
ft. X 10 ft.). It is crossed by three lines : one is 3 ft. 
from the eaves, and parallel to it; one is 4J ft. 




CONSTRUCTIVE DRAWING. 81 

from the right-hand gable end, and parallel to it, 
and one is oblique; one end of the plan of the 
oblique line is in the plan of the upper edge, 5 ft. 
from the right-hand end, and the other end is in 
the plan of the eave, 7 ft. from the left-hand end. 
Draw the plan and elevation of the roof and lines; 
scale, J in. to the foot. 

Prob. 3. — An equilateral triangular pyramid meas- 
ures 33 in. at the base of each side, and has a pitch 
of 48 in.; it is cut by a plane which cuts each side 
in a straight line. There are three sides, with one 
straight line in each side, and these lines meet on 
the hip (edges of the sides). The plans of the 
points on the hips are 4 in., 9 in., and 15 in. from 
the plan of the apex. Draw the plan and eleva- 
tion; scale, -jig- in. to the foot. 

Prob. 4.— A rectangular pyramid, 3 ft. square at 
the base and 9 ft. pitch, is cut by a plane. The plan 
of the points of intersection with the hips are 15 
in., 12 in., 9 in., and 12 in. from the plan of the 
apex. Draw the plan and elevation ; scale, 1 in. to 
the foot. 

Rule. — Draw the plan and elevation of the given 
inclined surface, and the plan of the given line on 
the surface. From the ends of the plan of the 
given line, draw leaders to intersect the elevations 
of the ed^es of the surface in which the ends of 
the given line rest. These points of intersection 
are the elevations of the ends of the given line. 
Join these points of intersection by a right line, 
and this line will be the required elevation of the 
given line. 



82 



MECHANICAL DRAWING. 



Section III. — To find the elevation of a point in an 
inclined surface when the plan and elevation of the 
surface and the plan of the point are given. 

Prob. 1.— In one side of a gable roof 27 ft. long, 
,—______ 14 ft. span, and 6 ft. pitch, is a 

/ p /\ hole, and the plan of this hole is 
; ^ 13 ft. from the plan of the left- 
hand gable end and 4 ft. from the 
plan of the eave. It is required to find the eleva- 
tion of the given point; scale, ^ in. to the foot. 
(Fig. 104.) 

Solution.— Draw the plan abed, and locate the 
plan of the given point, p. c 

Through p draw any line, ef, 
crossing the plan of the side of 
the roof. Find the elevation of 
the roof and the line ef. From 
p draw a leader to intersect e'f 
in ]/, and p' is the required eleva- 
tion of the given point. (Fig. 
105.) 

Prob. 2.— Draw the plan and 
elevation of a rectangular pyra- 
mid, 9 in. X 12 in. base, and 8 in. 
pitch, pierced by a line parallel 
to the axis ; the plan of the point where this line 
pierces one of the larger of the inclined surfaces of 
the pyramid is 1J in. from the edge of the plan of 
the base, and 5 in. from the left-hand edge of the 
plan of the face that it pierces ; scale, § in. to the 
foot. 




CONSTR UCTIVE DRA WING. 83 

Prob. 3. — A coppersmith is required to construct 
an equilateral triangular prism, each face of which 
is 8 in. wide and measures 13 in. parallel to the 
axis. Through this prism is to pass a fine straight 
wire, perpendicular to one of its faces, which it 
pierces 2 in. from one edge and 4 in. from the end. 
Draw plan and elevation of the prism and the holes 
for the wire; scale, J in. to the inch. 

Rule.— Draw the plan and angular elevation of 
the surface ; locate the given plan of the point, and 
through it draw any line crossing the plan of the 
surface. Find the elevation of a line on the given 
surface of which this line is the plan. Draw a 
leader from the plan of the point to intersect this 
line, and the point of intersection will be the re- 
quired elevation of the point. 

Section IV. — To find the elevation of a point in an 
inclined surface by parallel elevations on the same 
base when the plan of the point and the plan and 
elevation of the surface are given. 

Prob. 1. — Draw the plan and parallel elevations 
of the south and east sides of a gable-roofed house, 
standing with its longer sides north and south ; 16 
ft. wide, 20 ft, long, and 10 ft. high ; 1 ft. founda- 
tion ; roof pitch, 9 ft., and projection, 1 ft. Plan 
of chimney, 2 ft. square, 3 ft. 6 in. from the east 
wall, and 11 ft. from the north wall ; chimney 
projection above the roof, 6 ft.; scale, 4± in. to the 
foot. 



84 



MECHANICAL DRAWING. 



d; 







r 



Fig. 106. 



Solution. — Draw the plan, abcdmnef, and the end 

elevation, a'b'. 
This gives us 
the height of 
the point where 
the chimney 
comes out of 
the roof. Re- 
« volve the side 
be, a n d con- 
struct a side 
elevation of the 
house and 
chimney. The 
elevation of the 
point where the chimney will come through the 
roof is located in e n, f n . 

Prob. 2. — A solid rectangular pyramid, 4 in. X 6 
in. base and 9 in. pitch, is pierced by a one inch 
square hole, perpendicular to its base, and cutting 
it midway between its shorter edges, and midway 
between the plan of the apex and the plan of the 
base; scale, J in. to the inch. 

Prob. 3. — Draw a plan and angular elevation 
of a hexagonal pyramid, 12 in. diameter X 18 in. 
long, lying on its side, and pierced by a vertical line 
which passes through it 4J in. from the axis and 7 
in. from the end; scale, \ in. to the inch. 

Prob. 4. — The sides of the base of a triangular 
pyramid are 1} in., \\ in., and 2 in., and the pitch 
is 4| in.; each of its sides is pierced by a vertical 
line. The plan of one of these lines is \ in. from 
the base and \ in. from the right-hand edge of the 



CONSTRUCTIVE DRAWING. 85 

plan of the longest face; the plan of the second line 
is J- in. from the shortest base, and 1 in. from the 
left-hand edge of the plan of the smallest face; and 
the plan of the third is T 3 g- in. from the base and f 
in. from the plan of the right-hand edge of the 
third face. Draw the plan and the elevation ; half 
size. 



Rule. — Draw the plan of the given surface, and 
locate the given plan of the required point. From 
this plan construct an end elevation of the given 
surface and point, and from these construct a 
parallel side elevation on the same base-line as the 
end elevation. The three drawings so constructed 
will completely describe the given surface and the 
required point. 



CHAPTER V.— The Description of Points in a 
Surface of Eegular Curvature. 

A Surface of Regular Curvature follows a fixed 
law. 

Remark. — A sphere has a surface of regular cur- 
vature, because, having any point on its surface and 
the center of the sphere given, its entire surface 
may be exactly determined and described. 

An egg has a surface of irregular curvature, be- 
cause, having any number of points given, it would 
still be impossible to determine exactly and de- 
scribe the remaining points on its surface. 



86 



MECHANICAL DRA WIXG. 



Section I. — To find a point on a regular curved sur- 
face when the plan and elevation of the surface 
and the plan of the point are given. 

Prob. 1. — A hemisphere 2 in. in diameter has a 
point on its surface, and the plan of this point is f 
in. from the center of the plan of the sphere. 
Draw the plan and elevation of the hemisphere 
and point; scale, ■£ full size. 

Solution. — Draw the plan and elevation of the 
hemisphere ab and a'b'd' '. (Fig. 
107.) As the surface of a 
sphere may be regarded as 
consisting of any number of 
parallel circles increasing and 
diminishing in diameter ac- 
cording to a fixed law, every 
point on the surface of a 
hemisphere may be regarded 
as located on a circle parallel 
to its base. Locate the point 
p § in. from the center of 
the plan, and through it de- 
scribe a circle, ef parallel 
to ab. This circle, ef is the 
plan of a circle on the surface 
of the hemisphere. To find 
the elevation of ef draw lead- 
ers tangent to it, to intersect 

the outline of the elevation in e' and f. Join e! 

and /', and e'f is the elevation of a circle on the 




Fig. 107. 



CONSTRUCTIVE DRAWING. 



87 



surface of the hemisphere which contains the given 
point. Draw the leader, pp f , to intersect e?f, and p' 
is the required elevation of the given point, which 
is now completely described. 

Prob. 2. — Draw the plan and elevation of a cone ; 
base, 46 ft. in diameter ; and pitch, 63 ft., with a 
point on its surface, of which the plan is 13 ft. 
from the center of the plan of the cone ; scale, -^|- 
ino to the foot. 



Solution. — Draw the plan and elevation of the 
cone, abc and a'b'c'. As 
the surface of the cone 
may be regarded as 
made up of a series of 
circles parallel to each 
other, and increasing 
and diminishing accord- 
ing to a fixed law, the 
point on the surface of 
this cone may be con- 
sidered as being on one 
of these circles, the plan 
of which is 13 ft. radius. 
Draw the plan of this circle, ef y and from this draw 
leaders tangent to it to meet the elevation in e' and 
f. Draw c'f, and this will be the elevation of the 
circle containing the elevation of the point given. 
Locate the plan of the point p, and from it draw a 
leader to intersect e'f in p' ; p' is the elevation of 
the given point, and the point and the cone are 
completely described. 




Fig. 108. 



88 MECHANICAL DRAWING. 

Prob. 3. — Draw the plan and elevation of a hemi- 
sphere 10 ft. in diameter, with a point on its sur- 
face, the plan of which is 3 ft. from the center of 
the plan of the hemisphere; scale, J in. to the foot. 

Prob. 4. — Draw the plan and elevation of a cone, 
1 in. radius at the base and 2 J in. pitch, with a 
point on its surface, the plan of which is f in. from 
the center of the plan of the cone ; scale, half size. 

Prob. 5. — Draw the plan and elevation of a sphere 
5 ft. in diameter, having two points on its surface, 
one above and one below the center. The plan of 
the point above the center is 1 ft., and the plan of 
the other is 9 in., from the plan of the center of 
the sphere ; scale, half size. 

Prob. 6. — A cone, 10 ft. in diameter at the base 
and 15 ft. pitch, is pierced by four horizontal lines 
12 ft, long. The plan of the point where the sur- 
face of the cone is pierced by one of the lines is 1 
ft. from the center of the plan of the cone ; the 
plan of another is 2 ft. from the plan of the center; 
the plan of the third is 3 ft. from the plan of the 
center, and the plan of the fourth is 4 ft. from the 
plan of the center. All the lines project equally, 
and their plans form equal angles with each other, 
and meet on the vertical axis of the sphere. Draw 
the plan and elevation; scale, J in. to the foot. 

Prob. 7. — A hemisphere, 33 ft. radius, is pierced 
in each quadrant by a line that is 20 ft. in length in 
the plan, and passes through the center. The plan 
of a point where one of these lines pierces the sur- 
face of the hemisphere is 5 ft. from the center of 
the plan and 6 ft. from the right-hand side of the 
first quadrant; the plan of another point is 14 ft. 



CONSTRUCTIVE DRAWING. 



89 



from the center and 7 ft. from the left-hand side of 
second quadrant; the plan of the third point is 9 ft. 
from the center of the plan and 3 J ft. from the 
right-hand side of third quadrant ; and the plan of 
the fourth point is 11 ft. from the center and 4 ft. 
from the left-hand side of fourth quadrant. Draw 
the plan and elevation ; scale, -^ in. to the foot. 



Section II. — To find the plan of a point on a surface 
of regular curvature when the plan and elevation 
of the surface and the elevation of the point are 
given. 

Prob. 1. — Draw the plan and elevation of a hemi- 
sphere 3 in. in diameter, and having a point on its 
surface, of which the elevation is 1J in. from the 
hase-line and 1J in. measured horizontally from the 
right-hand edge of the elevation of the hemisphere; 
scale, I full size. (Fig. 109.) 

Solution. — Draw the plan and 
elevation of the hemisphere, abd 
and a'b'g, and locate the point p r 
in the elevation. Through the 
point p' draw e'f parallel to the 
hase, and it will be the elevation 
of a circle passing through the 
given point on the surface of the 
hemisphere. From e r and f draw 
leaders, and with c as a center de- 
scribe the plan of the circle, efi 
tangent to the leaders. Draw the 
leader pp' to meet this circle, ef ; p is the required 




90 MECHANICAL DRA WING. 

plan of the given point, and the hemisphere and 
the point on its surface are completely described. 

Prob. 2.— A hemisphere 25 ft. in diameter has a 
point on its surface, the elevation of which is 2 ft. 
high and 6 ft. measured horizontally to the left- 
hand of the vertical axis. Draw the plan and 
elevation of hemisphere and point; scale, J- in. to 
the foot. 

Prob. 3. — A cone, 3 in. in diameter at the base 
and 4J in. pitch, is pierced by a vertical line at a 
point on the surface of the cone one third of its 
height above the base. Draw the plan and eleva- 
tion of the cone and line; scale, J in. to the inch. 

Prob. 4. — A sphere 12 in. in diameter is pierced 
by two sets of three horizontal lines, each 15 in. in 
length, and the lines of each set form etpial angles 
and meet on the vertical diameter of the sphere. 
One set of lines is 2 in. above, and the other is 4} 
in. below, the center of the sphere. Draw the plan 
and elevation of the sphere and the lines; scale, \ 
in. to the inch. 

Prob. 5. — A cone, 100 ft. in diameter at the base 
and 150 ft. pitch, is pierced by live lines, passing 
through the center of the base and piercing the 
surface of the cone. The elevations of the points 
on the surface of the cone are located as follows : 
24 ft. high, and 40 ft. to the left-hand of the axis; 
12 ft. high, and 10 ft. to the right-hand of the 
axis ; 40 ft. high, and 36 ft. to the left-hand of the 
axis on one side; 90 ft. high, and in the middle; and 
75 ft. hiffh, 28 ft. to the left-hand of the axis on 
the other side. Draw the plan and elevation of 
the cone and the lines; scale, -fa in. to the foot. 



CONSTR UCTIVE DRA WING. 



91 



CHAPTER VI. — The Intersection of Horizontal 
Lines with Inclined Surfaces. 

Prob. 1. — Draw the plan and elevation of a house 
16 ft. w r ide, 20 ft. long, and 18 ft. high, including 
foundation, 2 ft.; gable roof, 10 ft. pitch, 16 ft. span, 
and projecting 1 ft.; dormer window, with gable 
roof in middle of side of roof, 4 ft. wide, 8 ft. high 
to the ridge of the 
roof, 2 ft. pitch, 
projecting 6 in., 
and the face flush 
with the wall of 
the house; scale, 
J ¥ in. to the foot. 

Solution. — Draw 

the end elevation 

of a house and the 

side elevation of a 

dormer window, 

a' b' o' p ! rn f a' . From 

this construct the 

plan, and find p, 

which is the plan of the point where the ridge of 
the dormer window, op, will meet 
the roof. From these construct the 
side elevation. (Fig. 110.) 

Prob. 2.— A house is 18 ft. X 28 
ft. X 20 ft, high, with a gable roof, 
9 ft. pitch, and a wing 16 ft. X 3 ft. 

projection, and 16 ft. high; eaves and gable project 

1 ft. (Fig. 111.) Problem: to find where the ridge 

M. D.— 8. 




Fig. 110. 




Fig. 111. 



92 



MECHANICAL DRAWING. 




Fig. 112. 



of the wing will meet the roof of main house; 
scale, J in. to the foot. 

Prob. 3. — A house is 24 ft. square, 18 ft. high, 
with a gable roof 8 ft. pitch; it has 
an extension 12 ft. square and 12 
ft. high, projecting 4 ft. beyond, 
and extending 8 ft. along the side 
of the main house. Make a work- 
ing drawing ; scale, J in. to the 
foot. 

Def. — A working drawing is a complete descrip- 
tion of all the dimensions of an object by plans 
and elevations. 

Prob. 4.— A barn is 18 ft. X 24 ft., 16 ft. high, 

gable roof f pitch (J span), with a 

dormer window 5 ft. wide and 5 ft. 

high, having a gable roof, f pitch, 

7 ft. from the end of the barn. 

Make a working drawing; scale, J 

in. to the foot. " (Fig. 113.) Fig - 113 - 

Prob. 5.— A house" is 24 ft. X 36 ft., 20 ft. high, i 
pitch, gable roof. There is a pro- 
jection 3 ft. X 18 ft., 6 ft. from 
right-hand end of the longer side, 
having a gable roof J pitch ; and 
a dormer window, 5 ft. wide and 
h pitch, 4 ft. from the left-hand 
end of the roof and 6 ft. from the 

eaves. Make a working drawing; scale, J in. to 

the foot, (Fig. 114.) 

Rule. — Draw the end elevation of the inclined 
surface and the side elevation of the given line. 




Fig. 114. 



CONSTR UCTIVE DRA WING. 



93 



From this, draw the plan of the surface and line. 
From these draw the side elevation of the surface 
and the end elevation of the line. By the plan and 
elevations so drawn, the line will be completely 
described. 



Miscellaneous Problems. 




Fig. 115. 



Prob. 1— A house is 24 ft. X 40 ft., 24 ft. high, 
and J pitch gable roof (12 ft.). In the middle of 
one of the longer sides is a rectangular oriel win- 
dow, 10 ft. wide, 9 ft. high, and projecting 3 ft; 
with gable roof one half pitch (5 ft.), and eaves con- 
tinuous with eaves of main house. 
There is a hexagonal oriel window 
at one corner, 8 ft. in diameter 
and 9 ft. high, showing 5 faces; 
this window has a one half pitch 
(4 ft.) gable roof, and its eaves are 
continuous with the eaves of the 
main house. Make a ground plan, a second story, 
and a roof plan, and an end and two side eleva- 
tions; scale, J in. to the foot. 
(Fig. 115.) 

p ro b. 2.— A house is 36 ft. in 
diameter; it is 15 ft. high, and 
has a hemispherical roof; there 
is a wing 14 ft, X 16 ft., 15 ft. 
high, with a half pitch roof. 
Make a plan, a side, and an end elevation; scale, J 
in. to the foot. (Fig. 116.) 




Fig. 116. 



94 



MECHANICAL DRAWING. 



Prob. 3. — A house is 24 ft. square, 18 ft. high, 
and has a hip roof, f pitch; 
it has a wing 14 ft. square 
and 18 ft. high, 2 ft. from 
the left-hand corner of the 
house with a half pitch roof; 
a dormer window 4 ft. wide 
and 3 ft. high, with a lean-to 
roof, 3 ft. pitch, and the sill 
Draw a 




Fig. 117. 

3 ft. from the eaves 
ground plan, a second floor plan, a 
roof plan, and a front and a side 
elevation; scale, J in. to the foot. 
(Fig. 117.) 

Prob. 4. — Draw plans and eleva- 
tions of a house with a roof similar 
to that described by Fig. 118. 



Fig. 118. 



CHAPTER VII.— Conic Sections. 



Prob. 1. — Draw the plan and elevation of a frus- 
tum of a cone 5 in. in diameter at the base, 8 in. 
pitch, and cut off at the top by a plane cutting the 
axis 4 J in. from the base at an angle of 45° ; scale, 
\ in. to the inch. 

Solution. — Draw the plan and elevation of the 
cone, (tbc and a'b'c', and the end elevation, win 1 , of 
the cutting plane, and a'b'n'm' is the elevation of 
the required frustum. It is required now to find 
the plan of the frustum where it is cut off by the 



CONSTRUCTIVE DRAWING. 



95 





Fig. 119. 



Fig. 120. 



plane m'n'. Draw the elevations and plans of any 
number of circles, 1-5, on 
the surface of the cone that 
will be cut by the plane 
m'n' and the points 1', 2', 3', 
4', 5, m', and n' will be the 
elevations of ten points in 
the top of the frustum. 
Find the plans of these 
points, and join them by a 
curved line which will de- 
scribe the plan of the top 
of the required frustum. 
(Figs. 119, 120, 121.) 

Prob. 2. — Draw a plan 
and elevation of a cone 13 
ft. in diameter at the base, 
25 ft. pitch, and cut by a 
plane making an angle of 
60° with the axis, 15 ft. 
from the base; scale, -^ in. 
to the foot. 

Prob. 3. — Draw a plan and 
elevation of a frustum of a 
cone 4J in. in diameter at 
the base, 5 in. pitch, and 
the plane of the top making 
an angle of 45° with the 
axis, 2J in. from the base ; Fig. 121. 

scale, f in. to the foot. 

Prob. 4. — Draw a plan and elevation of a frustum 
of a rectangular pyramid, base 63 ft. square, pitch 
105 ft., angle of the top 90° with one of the sides, 




96 MECHANICAL DRAWING. 

and length of axis 80 ft.; scale, ^ in. to the 
foot. 

Prob. 5. — Draw the plan and elevation of a frus- 
tum of a hexagonal pyramid 2 in. in diameter at 
the base, 4 in. pitch, angle of the top 45° with one 
of the hips (angles), and the axis 3 in. in length; 
scale, full size. 

Prob. 6. — Draw the plan and elevation of a frus- 
tum of an oblique cone, 23 ft. in diameter at the 
base ; angle of the axis with base, 60° ; length of 
axis, 15 ft.; top, 4 ft. radius, and parallel to base; 
scale, J in. to the foot, 

Prob. 7.— Draw the plan and elevation of a frus- 
tum of an oblique cone, 10 ft. in diameter at the 
base, axis inclined 60° from the vertical and 5 ft. 
long, upper face perpendicular to axis, and 5 ft. 
wide in its longest diameter ; scale, i in. to the foot. 



CHAPTER VIII.-The Helix. 

A Helix is a spiral line that continually advances 
in the direction of a straight line called its axis. 

If the spiral line ascends and passes from left to 
right in front of the axis, when the axis is vertical, 
the helix is said to be right-hand. 

If the spiral line ascends and passes from right to 
left in front of the axis, when the axis is vertical, the 
helix is said to be left-hand. 

The Pitch of a helix is the distance it advances 
in making one revolution, and is measured parallel 
to the axis. 



CONSTRUCTIVE DRAWING. 



97 



Section I. — Helical Lines. 

Prob. 1. — Draw the plan and elevation of a right- 
hand helical line 1 in. in diameter, 2J in. long, and 
1J in. pitch. 



Solution. — As the required 
helix is 1 in. in diameter 
and 2| in. in length, draw 
the plan and elevation of a 
cylinder having those dimen- 
sions, acb and a'c'c"a" (Fig. 
122). 

Since the 

aw 
from a'c'. 

As this 
from a r to 



pitch 



is 



li 



in., 

in. 



helix advances 
"' in making one 



revolution, it will advance -^ 
of that distance in making 
-^ of one revolution. 

This heinsf" a riffht-hand 



heing a 
helix, it will pass from left 
to right. Begin at a and a', 
and divide the plan, abc, and 
the pitch, a'a'", into 12 equal 
divisions, 1-2 to 12. Through 
these division points in the ele- 
vation, draw dotted horizontal 
lines, and from the division 
points in the plan draw 
leading lines to intersect them 




Fig. 122. 



Join these points 



98 



MECHANICAL DRAWING. 



of intersection by a curved line. This line will be 
one pitch (1J in.) of the required helix. The re- 
maining portion (1 in.) may be described in the 
same way. 

Note. — To describe a left-hand helix, the line would pass 
from right to left instead of from left to right, as in the last 
problem ; that is, in describing a left-hand helix, a and c would 
exchange places. 

Prob. 2. — Make a constructive drawing of a right- 
hand helical line, 1 ft. pitch, 1 ft. diameter, and 2 
ft. long; scale, 2 in. to the foot. 

Prob. 3. — Make a constructive drawing of a left- 
hand helical line, 6 in. pitch, 3 in. diameter, and 9 
in. long; scale, J in. to the foot. 

Section II. — Helical Bands. 

A Helical Band is the space 
between two similar and par- 
allel helices. 

Prob. 1. — Make a constructive 
drawing for a right-hand hel- 
ical band J in. wide, 1 in. in di- 
'&£ anieter, f in. pitch, and 111 in. 
long. 

Solution.— Draw plan and 
$"j: P-if elevation of a cylinder 1 in. in 

diameter and 111 i n . long. (Fig. 
123.) Construct a right-hand 
helix, | in. pitch. Beginning 
i in. from the bottom of the 
elevation, draw another right-hand helix having the 




Fig. 123. 



CONSTRUCTIVE DRAWING. 



99 



same pitch as the first. The space between these 
helices will be the helical band required in the 
problem. 

Prob. 2. — Make a constructive drawing for a 
right-hand helical band i in. wide, 2 in. long, 2 in. 
in diameter, li in. pitch. 

Prob. 3. — Make a constructive drawing for a left- 
hand helical band f in. wide, 2 J in. long, If in. 
radius, and If in. pitch. 

Prob. 4. — Make a constructive drawing for a left- 
hand helical band 10 ft. wide, 43 ft. long, 28 ft. 
diameter and 36 ft. pitch ; scale, -^ in. to the foot. 

Prob. 5. — Make a constructive drawing for a right- 
hand and a left-hand helical band, \ in. wide, cross- 
ing each other. Diameter 2 in., ^^^_^^ 
length 2 in., pitch 2 in. /l I <\ 

Section III. — Helical Flanges. y~~~%Y~\~^'y~<l 

A Helical Flange is a surface !\Ji ; | i \\ b /\\ 

that is between two parallel hel- ;_]_ iH^^oHi 

ices, and is perpendicular to the Lji^rr-SSS^H 

axis of the helix. rczji^S^&SSi 7 

Prob. 1. — Make a constructive S^S^^pTiri-i 

drawing for a right-hand helical 1 R|^^--ji : J:1c : -j : i 

flange J in. wide, 1 in. outer tjrjrljj^^i^^ 

diameter, 1^- in. long, and f in. l Ss:^iD~.i~J^^ 7 

pitch. fe|^^^^ 

Solution. — Draw plan and ele- «-:2fc-.r.7::_;: 

vation for a right-hand helix, 1 Fig * 124 ' 

in. diameter, 1^- in. long, and f in. pitch, 1-4-7-10-12 

(Fig. 124). Within this helix, and on the same 

axis, draw the plan and elevation of another right- 

M. D.— 9. 



100 



MECHANICAL DRAWING. 



hand helix J in. diameter, 1^- in. long, and j in. 
pitch, a-d-g-j. The space between these helices rep- 
resents the flange required in the problem. 

Prob. 2. — Make a constructive drawing for a 
right-hand helical flange J in. wide, 2 in. inner 
diameter, 3 in. pitch, and 4 in. long. 

Prob. 3. — Make a constructive drawing for a left- 
hand helical flange 2J in. in diameter, 1J in. pitch, 
1 in. wide, and 3 in. long. 

Prob. 4. — A cylinder 2 in. in diameter and 4 J in. 
long, has on its surface a right-hand helical flange 
1J in. pitch and J in. wide. Make a constructive 
drawing; scale, J in. to the inch. 

Section IV. — Angular Helical Projections. 

Prob. 1. — Make a con- 
structive drawing for a 
screw 1J in. in diameter, 
1J in. long, f in. pitch ; 
V thread, J in. deep and 



Solution. — Draw the 
plan and elevation of a 
helical band, J in. wide, 
| in. pitch, and 1 in. di- 
ameter and 1J in. long, 
I- VI and a-g (Fig. 125). 
Draw the helix 1-7, be- 
ginning -^ in. from bot- 
Fig - 125> torn of elevation, 1J in. 

diameter, and f in. pitch. Connect these helices by 
straight lines, and the screw is described. 




CONSTRUCTIVE DRAWING. 



101 



Prob. 2. — Make a constructive drawing for a 
right-hand Y thread screw, 2 J in. long, 4| in. diam- 
eter, 2 in. pitch, and threads J in. deep and If in. 
wide. 

Prob. 3. — Make a constructive drawing for a left- 
hand Y thread screw 3 in. long, 1J in. in diameter, 
1J in. pitch ; threads -^ in. deep and 1J in. wide. 

Prob. 4.— Make a constructive drawing for a 3J 
in. right-hand Y thread screw, 1 in. long, 1J in. 
pitch ; threads meet at bottom of groove, the depth 
of which is equal to J the pitch. 

Prob. 5. — Make a constructive drawing for a left- 
hand Y thread screw 4 in. in diameter, 3 in. long; 
angle of thread 87J°, and 
pitch 



1J in. 



Section V. — Rectangular 
Helical Projections. 

Prob. 1. — Make a con- 
structive drawing for a 
right-hand, rectangular, 
helical projection 1J in. 
in diameter, If in. pitch, 
and i in. square, on a 
cylinder If in. long. 

Solution. — Draw the 
band, 1-7 and I -VII 
(Fig. 126), i in. wide, 1J 
in. in diameter, and If 
in. pitch. Draw the 
cylinder If in. long and Fig. 126. 

1 in. in diameter. Draw the parts of the helices, 




102 



MECHANICAL DRAWING. 



abed and efg, that show at the upper and lower 
sides of the helical projection. Draw the exposed 
parts of the cylinder, and the conditions of the 
problem are completely satisfied. 

Note. — The helical projection described in Fig. 126 is called 
a square thread. 

Prob. 2. — Make a constructive drawing for a left- 
hand, square thread screw, 3 in. long, 1J in. pitch, 
2J in. diameter, and thread f in. square. 

Prob. 3. — Make a constructive drawing for a 
right-hand, square thread screw, 16 in. long, 24 in. 

in diameter, 6 in. pitch; 
thread, J of the pitch ; 
scale, Jg- in. to the inch. 
(Fig. 127.) 

Prob. 4. — Make a con- 
structive drawing for left- 
hand, square thread screw 
2 in. long, 1J in. diameter, 
1J in. pitch. 

Prob. 5. — Make a con- 
structive drawing from 
the object for a square 
thread screw with head, 
nut, and washer, full size. 
Prob. 6. — Make a con- 
structive drawing for a 
flight of 16 winding stairs, 
inner radius 1 ft. and outer radius 5| ft. ; riser, 7J 
in. Plan of stairs a complete circle, and each step 
a single stone block setting over 1J in. on the step 
below it; scale, \ in. to the foot. 




Fig. 127. 



CONSTR UCTIVE DRA WING. 103 

Prob. 7. — Make a constructive drawing for a 
right-hand hand-rail 6 in. wide, 3 in. thick, 10 ft. 
pitch, 5 ft. radius, and starting 3 ft. from the floor. 

Prob. 8. — Make a constructive drawing from the 
object for a flight of stairs with winders and balus- 
trade. 

Prob. 9. — Make a constructive drawing for a cir- 
cular staircase tower 25 ft. outside diameter, 39 ft. 
high, and wall 18 in. thick, pierced by 16 rectan- 
gular windows, 2 ft. wide and 8 ft. high, equally 
distant apart, and arranged in a helical line, mak- 
ing one revolution of the tower, and having a 20 ft. 
pitch, beginning 4 ft. from the ground; scale, J in. 
to the foot. 



CHAPTER IX.— Section Drawing. 

A Section Drawing describes the dimensions of 
an object where it is cut by a given plane. 

Prob. 1.— A cylinder 10 ft. long, 10 ft. in diame- 
ter, and 3 ft. bore, and standing with its axis 
vertical, is cut by three planes ; one of these planes 
is horizontal, and cuts the cylinder 5 ft. from the 
bottom; one plane is vertical, and passes through 
the axis ; and one is oblique, and makes an angle 
of 45° with the axis, cutting it 3 ft. from the base. 
Make a constructive drawing for the cylinder and 
each section ; scale, -^ in. to the foot. 

Solution. — Draw the plan and elevation of the 
cylinder, ab and a'b'a'b'. Draw the line EF for the 



104 



MECHANICAL DRAWING. 



elevation of the horizontal cutting plane, and per- 
pendicular to it, draw leaders from the points where 
it crosses the different parts of the elevation, and 
between these leaders construct the cross section 
drawing e'f. (Fig. 128.) Construct the section 
a"b r, a"b ,r , showing the parts cut by the plane AB. 
Draw the line CD, which will be the elevation of a 
plane cutting the axis of the cylinder at an angle 




Fig. 128. 



of 45°, and -^ in. from the base. From the points 
where this line crosses the different lines of the ele- 
vation of the cylinder draw perpendicular leaders, 
and perpendicular to these draw an axis, cd. Lay 
off on these leaders on each side of the axis the 
widths of the cylinder at the points from which the 



CONSTR UCTI VE DRA WING. 105 

leaders are drawn. Draw the outline of the section 
through these points, and shade that part that is 
cut by the plane. The several drawings so made 
will completely satisfy the requirements of the 
problem. 

Prob. 2. — Make a drawing of a section taken 
through the axis of a cylinder 12 in. in diameter, 
18 in. long, and 6 in. bore; scale, J in. to the inch. 

Prob. 3. — Make a drawing of a section taken 
3J in. from the center and parallel to the axis of a 
cylinder 14 in. in diameter, 5 in. bore, and 6 in. 
long; scale, J in. to the inch. 

Prob. 4. — Make a drawing of a section of a cone, 
9 ft. base and 1J ft. pitch, pierced parallel to its axis 
and 3 ft. from it by a hole 2J ft. in diameter, and cut 
by a plane passing through the center of the hole 
and the axis of the cone ; scale, J in. to the foot. 

Prob. 5. — Make a section drawing of a rectan- 
gular timber 2 ft. long, 3 J in. wide, and 9 in. thick, 
cut by a plane making an angle of 45° with the 
narrower edges ; scale, 1 in. to the foot. 

Prob. 6. — Make a section drawing of a cylinder 3 
ft. in diameter and 6 ft. long, cut through its center 
by a plane making an angle of 45° with the axis ; 
scale, 1 in. to the foot. 

Prob. 7. — Make a section drawing of a hollow 
cylinder 21 in. in diameter, 42 in. long, and 7 J in. 
bore, cut 18 in. from one end by a plane making 
an angle of 45° with the axis ; scale, 8 in. to the 
inch. 

Prob. 8. — A cubical block 8 in. square is pierced, 
perpendicularly to its faces, by three holes 2 in. in 
diameter, passing through its center. Make a 



106 MECHANICAL DRA WING. 

drawing of a section taken through two of its 
diameters. Make a drawing of another section 
taken through two of its diagonals; scale, J in. to 
the inch. 

Prob. 9. — Make a drawing of a vertical section 
taken through the center of the school-room ; scale, 
4 ft. to the inch. 

Prob. 10.— Draw a small two-story house in plan 
and elevation, and make a drawing of a vertical 
section taken through the center; scale, 8 ft. to the 
inch. 

Prob. 11. — Draw a common pump, and make a 
vertical section ; scale, J ft. to the inch. 

Prob. 12. — Procure a common spool. Make a 
plan and elevation, and an oblique section drawing ; 
scale, full size. 

Prob. 13. — A right circular cone is 5 in. in diam- 
eter at the base, and 11 in. high. Make a drawing 
of a section, cutting the axis 5 in. from the top at 
an angle of 45°; scale, 2 in. to the inch. 



CHAPTER X. — Foreshortened Dimensions. 

A Foreshortened Dimension is a dimension that 
is parallel to neither its plan nor its elevation. 

Prob. 1. — Make a constructive drawing of a tri- 
angular pyramid; sides of base, 3 ft., 2 ft., and 4 ft., 
and pitch 3 ft. 6 in.; plans of hips bisect the angles 
of the plan of the base; scale, J in. to the foot. 



CONSTRUCTIVE DRAWING. 



107 




Solution.— Let ABDC (Fig. 129) represent the 
pyramid pictorially, and it 
will be seen that each hip is 
the hypothenuse of a right- 
angled triangle, of which the 
other two sides are the plan 
and the pitch. The prob- 
lem, then, is to describe this 
triangle, and by so doing get 
a line that represents the required hip to the given 
scale. Draw the plan and elevation of the pyramid 

abed and a'b'cfd', and 
at one end of the plan 
of each hip draw a per- 
pendicular equal to the 
pitch. Join the end of 
this line with the other 
end of the plan of the 
hip. (Fig. 129, a). This 
line will be the hypoth- 
enuse of a right-angled 
triangle, of which the 
other sides are, respect- 
ively, the plan of the hip 
and the pitch of the 
pyramid. It, therefore, 
represents the true length of the hip, and aC" ', eC , 
and dC are the true lengths of the hips AC, BC, 
and DC represented to the scale of J in. to the foot. 
Prob. 2. — Make a constructive drawing which will 
describe the true length of the hips of a pyramid 
6 ft. square base, and 12 ft. pitch ; scale, J in. to 
the foot. 




Fig. 129, a. 



108 



MECHANICAL DRAWING. 




Fig. 130. 



Prob. 3. — What is the length of the hip of a 
pyramid 9 ft. X 15 ft. hase, 
piteh 12 ft., and ridge 6 ft. ; 
scale, i in. to the foot? (Fig. 
130.) • 

Prob. 4. — Make constructive 
drawings for an oblique pyra- 
mid with base 4 J ft. square, 
altitude 6 ft., and the plan of the apex 2 ft. out- 
side of the base, and opposite the middle of one 
side; scale, \ in. to the foot. 

Prob. 5. — Make constructive drawings for an oblique 
hexagonal frustum of a pyramid. Diameter of 
base, 21 in.; altitude, 27 in.; diameter of top, 13 in.; 
angle of axis with base, 60°; scale, J in. to the inch. 
Prob. 6.— Make constructive drawings for a frame- 
work 15 ft. square at base, 6 ft. square at top, 8 ft. 
high, outside measurement, and made of timbers 12 
in. square; scale, J in. to the foot. 

Prob. 7.— Make a drawing showing method of 
framing timbers in frame-work described in prob- 
lem 6; scale, 1 in. to the foot. 

Prob. 8. — A heavy piece of machinery has fallen 
into a cistern 3 ft. 6 in. in diameter, 
that opens into a room 11 ft. high. It 
is required to construct the largest pos- 
sible tripod crane with which to lift it 
out, the timbers of the crane to be 
8 in. X 12 in., and the feet of the crane 
must not be placed nearer than 1 ft. 
from the opening of the cistern. Make 
working drawings for crane; scale, \ in. to the foot. 
(Fig. 131.) 




Fig. 131. 



CONSTRUCTIVE DRAWING. 



109 



Rule. — Draw the plan and elevation of the given 
line. From one end of the plan draw a perpendic- 
ular equal to the pitch. Join the end of this line 
with the other end of the plan by a right line. 
This line will be the true length of the given line. 



CHAPTER XI. — Development of Surfaces. 

To Develop the Surface of any given object is to 
describe a diagram on some thin material, which, 
being cut out and bent or rolled into the proper 
shape, will inclose a space exactly equal and similar 
to that occupied by the given object. 



Section I. — Development of Plane Surfaces, 

Prob. 1. — Develop the surface of a rectangular 
box ii in. wide, ^f in. 
long, and ^f in. high. 

Solution.— Draw a 
plan of one side, abed, 
Fig. 132, if in. long, 
and li in. wide. Draw 
the elevation of each of 
the adjacent sides add" a", 
aa'Ub, bb"c"c, and dee'd'. 
Draw the plan of the 
remaining side of the 
box, d'c'W'a"'. Cut this 
diagram out and fold it 
on the appropriate lines, P ig# 132 . 













a'" 


h 






d' 


c' 




d'' 


d 


c 


C'/ 


a" 


a 


"b 


b" 




a' 


'0 





110 MECHANICAL DRAWING. 

and it will exactly envelop a rectangular space \^ 
i u# x |f in. X Jf in. It is, - therefore, the required 
developed surface of the given box. 

Prob. 2. — Develop the surface of a rectangular 
box 2 in. X 1J in. X 1 in. 

Prob. 3. — Develop the surface of a cube f in. 
square. 

Prob. 4. — Develop the surface of a square pyramid 
1} in. base and 3 in. pitch. 

Prob. 5. — A pan is f in. deep, the bottom is 2J 
in. X 3 in., and the sides flare J in. Develop the 
surface. 

Prob. 6. — Develop the surface of a frustum of a 
rectangular pyramid 3 in. square at the base and 6 
in. pitch; the plane of the top cutting the axis at 
an angle of 30°, 3 in. from the base. 

Note. — Pitch, as used in this book, always means completed 
height ; as, for instance, the pitch of a frustum is equal to the 
height of the completed object (cone or pyramid). 

Prob. 7. — Develop the surface of a hexagonal 
prism 2 in. long and f in. diameter. 

Prob. 8. — Develop the surface of a hexagonal 
pyramid 1J in. diameter at the base and f in. pitch. 

Rule. — Draw the plan of any face of the given 
object; and, joining the sides of this, draw the 
faces adjacent to it. Draw the face or faces adja- 
cent to these, and so on until all the faces have 
been described. The resulting diagram will be the 
required developed surface. 



CONSTRUCTIVE DRAWING. Ill 



Section II. — Development of single curved surfaces. 

A Single Curved Surface is a surface through 
any point of which one straight line can be drawn 
that will lie wholly within the surface. Cylindrical 
and conical surfaces are of single curvature. 

Cylindrical Surfaces. 

Prob. 1. — Develop the surface of a cylinder J in. 
long and ^ in. in diameter. 

Solution. — Suppose AB (Fig. 133) to be the 
given cylinder, and imagine that 
it has been freshly painted and *> f ._. ..jp^ l fr 
allowed to roll round once on some / | \ 

flat surface. The print of the / 2zdL___\ 

paint on this surface would de- 

r . Fig. 133. 

scribe the developed curved surface 
of the cylinder. The length of this print mark 
would be equal to the circumference of the cylinder 
and the width would be equal to the length of the 

cylinder. The circumference 
of the cylinder given in this 
problem is equal to -^ in. X 
3.1416 == .9812 in. == 1 in., and 
the width is J in. Draw the 
rectangle 1, 2, 3, 4, 1 in. long 
and } in. wide, and it will be 
the developed curved surface 
of the cylinder. (Fig. 134.) 
Draw two circles T 5 g in. in diameter, tangent to the 




112 



MECHANICAL DRAWING. 




sides of this rectangle, and the development is 
complete. 

Prob. 2. — Develop the surface of a cylinder 1| in. 
long and f in. in diameter. 

Prob. 3. — Develop the surface of a cylinder 3 in. 
long and f in. in diameter. 

Prob. 4. — Develop the surface of a cylinder 1^ in. 
long and J in. in diameter. 

Prob. 5. — A hollow cylinder, If in. in diameter 
outside, and | in. in diameter inside, and 1J in. long, 
is to be made of tin. Develop the surface inside 
and outside as well as at the ends. 

Prob. 6.— Develop the surface of a 
sheet-iron box that is J of a cylinder 2 
in. in diameter and 2 in. long. 

Prob. 7. — Develop the surface of a 
paper box 3 in. long, 2 in. wide, and 1 
in. high, with a half round cylindrical cover. (Fig. 
135.) 



Section III. — Cylinders, the ends 
of which are not parallel. 

Prob. 1. — Develop the surface 
of a cylinder J in. in diameter, f 
in. long at the axis, and cut off 
at one end at an angle of 45°. 

Solution. — Draw the plan, 1- 
12, J in. in diameter, and the 
elevation, l'-7', f in. long at the 
axis, and cut off at the top at an 
angle of 45°. (Fig. 136.) 



Fig. 135. 




CONSTRUCTIVE DRAWING. 



113 



Divide the plan into 12 equal parts. Draw lead- 
ers from these division points, and find the eleva- 
tions of the corresponding elements of the cylinder, 
2'-7'. 

Draw a line AB, equal in length to the circum- 
ference of the cylinder, 1 in. X 3.1416 = 3.1416 in.= 
3^- in. Divide this line into 12 equal parts, and at 
the division points erect perpendiculars equal in 
length to the elevations of the corresponding ele- 




ments of the cylinder. Join the tops of these lines, 
and the resulting diagram is the required developed 
surface of the given cylinder. (Fig. 137.) 

Prob. 2. — Develop the surface of one half a 
rectangular elbow of a round sheet-iron pipe, 2J in. 
long in the longest element, and 1 in. in diam- 
eter. 

Prob. 3. — Develop the surface of a 1J in. cylinder, 
cut off 1 in. from the base at an angle of 30° with 
the axis. 

Prob. 4. — Develop the surface of a If in. cylinder, 
li in. long, one end making an angle of 60° with 
the axis. 



114 MECHANICAL DRAWING. 

Prob. 5.— Develop the surface of one half of a 60° 
elbow, f in. diameter and 1| in. at its longest ele- 
ment. 

Prob. 6. — Develop the surface of one half of a 
150° elbow, | in. in diameter, and f in. at its 
shortest element. 

Section 1 V. — Development of conical surfaces. 

Prob. 1. — Develop the surface of a cone ; base, 1 \ 
in. in diameter and pitch ^ in. 

Solution.— Let bAc be the given cone lying on 
c _ its side and in 

a' "T\ 1 

{"""' \ I the process of 

/ \ / revolving. At 

\ / \ / ^ Je beginning 

/ a \ / of the re vol u- 

\/^^ ; / tion A was at a, 

\^ ( y^ and at the end 

"i> A will be at a'. 

Fig - 138 - Thenthelength 

of the arc a'ba will be exactly equal to the cir- 
cumference of the circle Ab, at the base of the 
cone, and ca, the radius with which it has been de- 
scribed, is exactly equal to the slant height of the 
cone. (Fig. 138.) 

Draw the plan and elevation of the cone, abc and 
a'b'c' (Fig. 139), having the dimensions given in the 
problem. From this ascertain the slant height, a f c\ 
and with AC = a'c' as a radius, describe the arc of 
a circle, ABA'. (Fig. 140.) 

The number of degrees that are to be laid off on 
this circle in order that the length of the arc 



CONSTRUCTIVE DRAWING. 



115 



shall be equal to the circumference 
the cone will bear the same 
relation to 360° as the radius 
of the base of the cone bears 
to its slant height. 

To ascertain this, divide 
the slant height of the cone 
1J in. by the radius of the 



of the base of 



base of the 


cone ^ 


in. « = 


9 






A 


8 _ 

9 


_ 8_ 1 

~ 16 2 


= 180°. 


Lay ofF 



on the arc ABA! 180°, and 
draw the radii AC and A'C. 
The inclosed sector is the de- 
veloped surface of the given 
cone. 





Fig. 140. 

Prob. 4. — Develop the surface of a 
and 2} in. pitch. 

Prob. 5. — Develop the surface of a 
frame 1 in. wide and 1J in. pitch ; 2 
in. long and half conical at the ends. 
(Fig. 141.) 

M. D.— 10. 



Fig. 139. 

Prob. 2. - De- 
velop the sur- 
face of a cone 
1J in. base and 
If in. pitch. 

Prob. 3.— De- 
velop the sur- 
face of a cone 
2 in. base and \ 
in. pitch, 
cone \ in. base 




Fig. 141. 



116 



MECHANICAL DRAWING. 



Rule. — Draw plan and elevation of cone, and de- 
scribe a circle having a radius equal to the slant 
height of the cone. Lay off on this circle an arc 
containing a number of degrees that have the same 
ratio to 360° that the radius of the base has to the 
slant height of the cone. Draw radii from the 
center to the ends of this arc, and the sector thus 
inclosed will be the developed surface of the cone. 



Section V. — To develop the surface of the frustum of 

a cone. 

Prob. 1. — The frustum of a cone has a base 1 in. 
in diameter; the length of the axis is ^J in.; the 
angle at the base is 70°, and the top is cut off at an 
angle of 45° with the axis. Describe the developed 
surface of the frustum. 

Solution. — Draw the plan and elevation, abc and 
a'b'c'i and describe the arc ABC, 
the length of which is equal to 
the circumference of the circle 
ab. Divide the plan into any 
number of equal parts, and draw 
radial lines to the center c. 
Draw elevations of these lines 
in c'1-7, c'2-6, and r'3-5. Divide 
ABA' into the same number of 
equal parts as the plan ab, and 
draw CI, C2—C1. From the 
points where the elevations of 
the elements meet the line a"b", 
the top of the frustum, draw 
Fig. 142. horizontal lines to meet c'U in 




yi|U 2,10 3|9 



L!ib 



CONSTRUCTIVE DRAWING. 117 

a", 1-7, 2-6, and 3-5. From C lay off the distances 
c'a", c'l, e'2 — c f 7, c'a". Join the points just found 
by a curved line, a"1234567a" and ABA'a"a", the 




inclosed figure will be the developed surface of the 
frustum described in the problem. (Figs. 142 and 
143.) 

Prob. 2. — Develop the surface of the frustum of a 
cone If in. base, If in. axis, 75° angle at the base, 
60° angle of top with axis. 

Prob. 3. — Develop the surface of the frustum of a 
cone 1J in. base, } in. axis, 30° angle at the base, 
40° angle of top with axis. 

Prob. 4. — Develop the surface of the frustum of a 
cone J in. base, 2 in. axis, 75° angle at base, 90° 
angle of top with element of cone. 

Prob. 5. — Develop the surface of the frustum of a 
cone 2 in. base, J in. axis, 22J° angle at the base, 
45° angle of top with element of cone. 

Section VI. — Development of surf aces of double curvature. 

Prob. 1. — Develop the surface of a sphere 1 in. in 
diameter. 



118 



MECHANICAL DRAWING. 



Solution. — Draw the plan and elevation of the 
sphere, abc and a'b'c'. (Fig. 144.) Divide one half 
of the circumference of the ele- 
vation into any number of equal 
divisions at the points II, III, 
IV, and through these points 
draw the elevations of horizontal 
circles. Find the plans of these 
circles. Divide the plan, abc, 
into any number of equal sec- 
tors, 1, 2 — 5, 6. Find the eleva- 
tions of the points of intersection 
of the sides of the section 1-2 
with the plans of the horizontal 
circles. Draw a line, the length 
of which is equal to one half 
the circumference of the sphere, 
AB, 1.57 in. Divide this line to 
correspond with the divisions of 
the elevation of the sphere, and 
through these points draw per- 
pendicular lines equal to the 
arcs of the circles at II, III, 
IV, V, VI, in the elevation. 
Join the ends of these lines by 
a curve, and the inclosed space 
will be the developed surface of one sixth of the 
whole surface of the sphere, which, being repro- 
duced six times, will completely envelop the 
sphere. 

Prob. 2. — Develop the surface of a sphere If in. 
in diameter. 




CONSTR UCTIVE DRA WING. 119 

Prob. 3.— Develop the surface of a sphere 2J in. 
in diameter. 

Prob. 4. — Develop the surface of a sphere J in. in 
radius. 

Prob. 5. — Develop the surface of a sphere If in. 
radius. 

Rule. — Draw plan and elevation of the sphere. 
Divide the circumference into any number of equal 
divisions. Through these division points draw ele- 
vations of horizontal circles. Draw the plans of 
these circles. Divide the plan of the sphere into 
any number of equal sectors, and find the eleva- 
tion of that sector that is parallel to the elevation. 
Draw a line equal in length to one half the circum- 
ference of the elevation, divide it to correspond, 
with the elevation, and through the division points 
draw perpendicular lines, the lengths of which are 
equal to the width of the sectors at the different 
points. Join the ends of these perpendiculars, and 
the inclosed figure will be the developed surface of 
one section of the given surface. 



CHAPTER XII. — Development of Oblique Con- 
ical Surfaces. 

An Oblique Cone is a cone in 
which the axis is not perpendicular sy 

to the base. (Fig. 145.) y/ / 

Prob. 1. — Develop the surface of an <^'"~sy 
oblique cone; base, ^ in. in diame- Fi s- 145 - 

ter, l^- in. pitch, and axis 45° with the base. 



120 



MECHANICAL DRAWING. 



Solution. — Draw a plan and elevation of the cone, 
and divide the plan of the base into any number 
of equal parts. Draw the plans and elevations of 
the elements of the cone at these points. These 
elements divide the surface of the cone into small 
triangles, the bases of which are equal portions of 
the base of the cone, and the sides are foreshort- 
ened dimensions. (Chap. X.) 




Fig. 147. 



To ascertain the length of the foreshortened 
sides of these triangles, draw the pitch line C'C" 
and extend the base of the cone beyond it. Lay 
off on this extended base from c" the lengths of the 
plans of the elements of cone, c"l, c"2-c"l. Join 
c'l, c'2-c'7, and these lines are the true lengths of 
the foreshortened sides of the required triangles. 
(Fig. 146.) 

Now draw a line CI equal to c'l, and with the 
end 1 as a center, describe arcs on both sides of 
CI, of which the radius is equal to -^ the circum- 
ference of the base of the cone. With C as a 



CONSTR UCTIVE DRA WING. 121 

center, and a radius equal to (72, describe arcs cut- 
ting these. There are now described two adjacent 
triangles on the surface of the cone. In the same 
manner describe the triangles adjacent to these 
until all have been described. (Fig. 147.) 

Prob. 2. — Develop the surface of a cone of which 
the base is If in. in diameter, pitch If in., and axis 
makes an angle of 60° with base. 

Prob. 3. — Develop the surface of an oblique cone; 
base, J in. radius; pitch, J in.; and 22J° axis with 
base. 

Prob. 4. — Develop the surface of a frustum of an 
oblique cone; base, 1 in. radius; pitch, 2 in.; axis, 
1 in. long, 45° with base. 

Prob. 5. — The top of a portable furnace 2 ft. 6 in. 
in diameter is 2 ft. 6 in. below the under side of 
the floor joists, and it is required to construct a 
conical top which will connect by a straight pipe 6 
in. long with a register box 12| in. X 19| in., of 
which the center is 13 in. outside of the furnace 
top, and the longer edges make an angle of 30° 
with the axis of the frustum. Develop the surface 
of the frustum. 



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